SB    27fi    Ib3 


-BOOK 

TUF. 

ERRESTRIAL  GLOBE; 


1 1 

!  \V  METHOD   OK 

1  ING    AND    OPERATING    GLOBES. 

| 


Bv    ELLEN    E.   FITZ. 


<M 
CNJ 


i  ON: 
GINN      AND      IIKATH. 


LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 


GIF^T  OF 


.""1.  ,    U}  '  Jllu/vuo  ( 

C/^ss     ? 


HAND-BOOK 

OF   THE 

TERRESTRIAL  GLOBE; 

OR, 

GUIDE  TO   FITZ'S   NEW  METHOD  OF  MOUNT- 
ING  AND   OPERATING   GLOBES. 

DESIGNED   FOR  THE  USE  OF  FAMILIES,    SCHOOLS,   AND  ACADEMIES. 

BY    ELLEN    E.  FITZ. 


-\. 

UNIVERSITY  ) 

or  & 

iroR^vJ^ 

GINN     AND     HEATH. 


BOSTON: 


1878. 


Copyright, 

BY  ELLEN  E.  FITZ. 
1876. 


Cambridge : 
Press  of  John   Wilson  and  Son. 


PREFACE. 


THE  following  pages  are  designed  solely  as  an  accom- 
paniment to  Fitz's  Globe.  They  therefore  contain 
only  the  text  which  is  thought  to  bear  expressly  upon 
the  use  of  this  globe.  A  brief  Appendix,  comprising 
an  historical  account  of  globes,  with  a  few  details 
respecting  their  construction,  forms  the  single  excep- 
tion. I  am  happy  to  render  acknowledgments  to  Prof. 
G.  A.  HILL  of  Harvard  College,  from  whom  I  have 
received  valuable  aid  in  the  preparation  of  the  book. 

As  regards  the  practical  teaching  of  the  subject,  it 
is  suggested  that  it  would  be  well  to  have  the  learner 
first  study  thoroughly  the  first  71  articles  of  Part  I.  ; 
then  take  the  Description  of  the  Globe  (in  connection, 
of  course,  with  the  Globe  itself) ;  and  then  proceed  to 
the  Problems,  going  back  to  the  remaining  articles 
of  Part  I.  when  he  comes  to  the  Problems  which  refer 
to  the  subjects  explained  in  those  articles. 

It  is  intended  in  a  future  edition  to  enlarge  this 
manual,  giving  it  more  the  form  of  an  Astronomy,  and 
having  a  portion  of  it  adapted  to  the  use  of  a  celes- 
tial globe. 

E.  E.  F. 
WATERTOWN,  January,  1876. 


a;> 


CONTENTS. 


PART      I. 

Page. 

SECTION     I.    GEOMETRICAL  DEFINITIONS 7 

SECTION  II.    GEOGRAPHICAL  DEFINITIONS 12 

SECTION  III.    ASTRONOMICAL  DEFINITIONS 15 

SECTION  IV.    PHENOMENA  CAUSED  BY  THE  MOTIONS  OF  THE 

EARTH 20 

THE  DIURNAL  COURSE  OF  THE  SUN    ....  37 

MEASUREMENT  OF  TIME 42 

THE  SEASONS 45 

TWILIGHT 50 

DESCRIPTION  OF  THE  GLOBE .    .  55 


PART      II. 


PROBLEMS. 

I.     To  find  the  Latitude  and  Longitude  of  a  given  Place     .       59 
II.     Given  the  Latitude  and  Longitude,  to  find  the  Place .     .       60 

III.  To  find  all  those  Places  that  are  in  the  same  Latitude 

or  Longitude  with  a  given  Place 60 

IV.  Given  two  Places,  to  find  their  Difference  of  Latitude    .       61 
V.     Given  two  Places,  to  find  their  Difference  of  Longitude,       61 

VI.     To  find  the  Antoeci,  Perireci,  and  Antipodes  of  a  given 

Place 62 

VII.     To  find  the  Distance  between  two  Places 63 

VIII.     Given  the  Latitude  or  Longitude  of  a  Place,  and  its  Dis- 
tance from  a  given  Place,  to  find  the  Place  whose 

Latitude  or  Longitude  is  given 64 

IX.     To  find  the  Length  of  a  Degree  of  Longitude  corre- 
sponding to  a  given  Latitude 65 


4  CONTENTS. 

X.     To  bring  a  given  Place  into  a  given  Hour  of  the  Day 

or  Night  .     .     .     .  • 66 

XI.     Given  the  Difference  of  Longitude  between  two  Places, 

to  find  their  Difference  of  Time 67 

XII.     Given  the  Difference  of  Time  between  two  Places,  to 

find  their  Difference  of  Longitude 68 

XIII.  Given  two  Places,  to  find  their  Difference  of  Time     .      69 

XIV.  Given  the  Hour  of  the  Day  at  a  Place,  to  find  all  those 

Places  on  the  Earth's  Surface  where  it  is  then  Noon, 

or  any  other  given  Hour 70 

XV.     Given  the  Hour  of  the  Day  at  a  Place,  to  find  the 

Hour  at  another  Place 72 

XVI.    To  represent  the  Three  Positions  of  the  Sphere, — 

Right,  Parallel,  and  Oblique 73 

XVII.    To  find  the  Times  of  Sunrise  and  Sunset  at  a  given 
Place  on  a  given  Day,  and  also  the  Lengths  of  the 

Day  and  of  the  Night 74 

XVIII.  To  find  the  Length  of  a  Period  of  Continuous  Day 
upon  a  given  Parallel  in  the  North  Frigid  Zone ; 
also  when  this  Period  begins  and  ends 76 

XIX.  To  find  the  Length  of  a  period  of  Continuous  Night 
upon  a  given  Parallel  in  the  North  Frigid  Zone ; 

also  when  this  Period  begins  and  ends 77 

XX.  To  find  the  Length  of  a  Period  of  Alternate  Day  and 
Night,  succeeding  one  of  Continuous  Day,  upon  a 
given  Parallel  in  the  North  Frigid  Zone ;  also  when 
this  Period  begins  and  ends 77 

XXI.  To  find  the  Length  of  a  Period  of  Alternate  Day  and 
Night,  succeeding  one  of  Continuous  Night,  upon  a 
given  Parallel  in  the  North  Frigid  Zone  ;  also  when 

this  Period  begins  and  ends 78 

XXII.     At  a  given  Place,  the  Day  being  given,  to  find  what 

other  Day  of  the  Year  is  of  the  same  Length  ...       80 

XXIII.  To  find  at  what  Latitude,  not  within  the  Polar  Circles, 

a  given  Day  is  of  a  given  Length 80 

XXIV.  At  a  given  Place,  the  Day  and  Hour  being  given,  to 

find  how  the  various  Regions  of  the  Earth  are  situ- 
ated with  regard  to  Day,  Night,  Twilight,  &c.  .  .  82 
XXV.  To  pass  the  Globe  through  the  four  Seasons  of  Spring, 
Summer,  Autumn,  and  Winter ;  also  to  show  how 
the  Northern  and  the  Southern  Hemispheres  are 
illuminated  upon  a  given  Day,  and  the  Sun's  Place 
in  the  Ecliptic 83 


CONTENTS.  5 

XXVI.  To  show  the  Rate  at  which  the  Sun  moves  between 
the  Tropics, — a  Rate  bearing  a  close  relation 
to  the  Rate  at  which  Day  or  Night  changes  in 
Length,  or  the  Rate  at  which  the  Sun's  Diurnal 

Arcs  increase  or  decrease  in  Extent 85 

XXVII.     The  Day  being  given,  to  find  the  Sun's  Longitude, 

Right  Ascension,  and  Declination 85 

XXVIII.     The  Day  being  given,  to  find  those  Places  where 

the  Sun  is  Vertical,  or  in  the  Zenith,  at  Noon   .      86 
XXIX.     The  Day,  Hour,  and  Place  being  given,  to  find 

where  the  Sun  is  Vertical 86 

XXX.     To  find  those  two  Days  of  the  Year  on  which  the 
Sun  will  be  Vertical  at  a  given  Place  in   the 

Torrid  Zone 87 

XXXI.  The  Length  of  the  Day  at  any  Place  being  given, 
to  find  the  Sun's  Declination,  and  the  two  Days 
of  the  Year  on  which  it  has  this  Declination  .  87 
XXXII.  A  Place  and  the  Day  of  the  Year  being  given,  to 
find  how  much  the  Sun's  Declination  must  vary 
to  make  the  Day  a  given  Time  longer  or  shorter 
than  the  given  Day  ;  also  how  many  Days  will 
elapse  during  the  change 89 

XXXIII.  To  find  in  what  Latitude  the  Sun  does  not  set 

during  a  given  Number  of  Days  in  the  Frigid 
Zone 92 

XXXIV.  Given  a  Day,  to  find  the  Equation  of  Time,  or  the 

Difference  between  Sun-Time  and  Clock-Time  .      92 
XXXV.     To  Describe  the  Course  of  the  Sun,  with  respect 
to  the  Horizon  and  visible  Heavens,  at  a  given 

Place  upon  a  given  Day 93 

XXXVI.     To  find  the  Sun's  Altitude  and  Azimuth  at  a  given 

Place,  the  Day  and  Hour  being  known      ...       95 
XXXVII.     To  find  the  Region  of  the  Earth  in  which  a  Solar 
Eclipse  is  visible,  the  Time  of  its  Occurrence  at 

Washington  being  given 96 

XXXVIII.     To  find  the  Region  of  the  Earth  in  which  a  Lunar 
Eclipse  is  visible,  the  Time  of  its  Occurrence  at 

Washington  being  given 96 

XXXIX.     A  Place  and  Day  of  the  Year  being  given,  to  find 

the  Length  of  Morning  and  Evening  Twilight    .       97 
XL.     A  Place  and  Day  of  the  Year  being  given,  to  find 
when  Morning  Twilight  begins,  and  Evening  Twi- 
light ends 98 


6  CONTENTS. 

XLI.     To  find  the  Length  of  a  Period  of  Twilight  lasting 
from  Sunset  to  Sunrise  at  a  given  Place  in  the  Tem- 
perate Zone  ;  also  when  this  Period  begins  and  ends,     100 
XLII.     To  find  the   Length  of  a  Period  of  Twilight  lasting 
from  Sunset  to  Sunrise  upon  a  given  Parallel  of  the 
Frigid  Zones ;  also  when  this  Period  begins  and  ends,     101 
XLIII.     To  find  the   Length  of  a  Period  of  Continuous  Twi- 
light  upon  a  given  Parallel  in  the   Frigid  Zones  ; 

also  when  this  Period  begins  and  ends 103 

XLIV.  To  find  the  Length  of  a  Period  of  Twilight  alternating 
every  twenty-four  hours  with  Dark-Night  upon  a 
given  Parallel  in  the  Frigid  Zones ;  also  when  this 

Period  begins  and  ends 104 

XLV.     To  find  the  Length  of  a  Period  of  Dark-Night  upon  a 
given  Parallel  in  the  Frigid  Zones ;  also  when  this 

Period  begins  and  ends 107 

MISCELLANEOUS  PROBLEMS 108 

QUESTIONS  FOR  EXAMINATION no 

APPENDIX 117 

INDEX 119 


ILLUSTRATIONS. 


MEASUREMENT  OF  ANGLES 8 

DIEDRAL  ANGLES 10 

GREAT  AND  SMALL  CIRCLES n 

MERIDIANS,  PARALLELS,  &c 15 

RIGHT  PARALLEL  AND  OBLIQUE  SPHERES 18 

ELLIPSE 23 

DIRECTION  OF  THE  EARTH  IN  ITS  ORBIT 24 

PARALLELISM  OF  THE  EARTH'S  Axis 26 

THE  SEASONS 31 

POLAR  AND  SIDEREAL  DAY 43 

OBLIQUITY  OF  SUN'S  RAYS 47 

TWILIGHT 52 


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UNIVERSITY  ] 
or  J 

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»i 


NEW     METHOD 


OF 


MOUNTING   AND    OPERATING    GLOBES. 


PART     I. 

SECTION   I. 

GEOMETRICAL   DEFINITIONS. 

1.  A  point   has   position,  without  length,  breadth,  or 
thickness. 

2.  A  line  has  length,  without  breadth  or  thickness. 

3.  A  straight  line  has  the  same  direction  between  any 
two  of  its  points. 

4.  A  curved  line  or  curve  changes  its  direction  at  every 
point. 

5.  A  surface  has  length  and  breadth,  without  thickness. 

6.  A  plane  is  a  surface,  such  that,  if  any  two  of  its  points 
be  joined  by  a  straight  line,  this  line  will  be  wholly  in  the 
surface.     No  surface  that  can  be  made  is  a  perfect  plane 
surface,  in  the  strict  mathematical  sense  of  the  term.     But 
the  surface  of  a  table,  if  carefully  made,  is  very  nearly  a 
plane ;  and  the  test  employed  by  the  maker  consists  in 
laying  a  straight  rule,  called  by  him  a  straight  edge,  on  the 
surface,  when  its  entire  length  ought  to  be  in  contact  with 
the  surface.     A  plane  corresponds  to  what  is  commonly 
known  as  a  fiat  surface. 

Every  surface  which  is  not  a  plane,  or  composed  of  plane 
surfaces,  is  a  curved  surface. 


8  NEW    METHOD    OF 

7.  A  circle  is  a  plane  bounded  by  a  curve,  called  its  cir- 
cumference, which  is  everywhere  equally  distant  from  a  point 
within,  called  the  centre. 

8.  The  radius  of  a  circle  is  a  straight  line  drawn  from 
the  centre  to  any  part  of  the  circumference. 

9.  The  diameter  of  a  circle  is  any  straight  line  passing 
through  the  centre,  and  terminated  on  both  sides  by  the 
circumference.     A  diameter  is  evidently  equal  to  two  radii. 

10.  An  arc  of  a  circle  is  any  portion  of  the  circumfer- 
ence.    An  arc  equal  to  one-fourth  of  the  circumference  is 
often  called  a  quadrant.     The  straight  line  joining  the  ex- 
tremities of  an  arc  is  called  the  chord  of  the  arc. 


Fig.  i. 


11.  An  angle  is  the  difference  of  direction  between  two 
lines  meeting  (or  tending  to  meet)  in  a  point,  called  the 
vertex  of  the  angle.  Thus,  in  the  figure,  the  lines  A  B 
and  C  D  form  an  angle  ;  these  lines,  if  prolonged,  meet  at 
the  vertex  O,  forming  the  angle  BOD. 

Angles  are  usually  measured  by  means  of  circular  arcs, 
the  vertex  of  the  angle  being  the  centre  of  the  circle.  It 
is  shown,  in  geometry,  that  angles  having  their  vertices  at 


MOUNTING  AND   OPERATING   GLOBES.  9 

the  centre  of  the  same,  or  of  equal  circles,  are  to  each 
other  as  the  arcs  intercepted  between  their  sides.  Thus, 
in  the  figure,  the  angles  BOD  and  BOH  are  to  each  other 
as  the  arcs  E  F  and  E  G.  In  order  to  express  conveniently 
the  values  of  arcs  as  the  measures  of  angles,  the  circum- 
ference of  a  circle  is  divided  into  360  equal  parts,  called  de- 
grees (marked  °),  each  degree  into  60  equal  parts,  called 
minutes  ('),  and  each  minute  into  60  equal  parts,  called 
seconds  (").  Then  the  number  of  degrees,  minutes,  and 
seconds,  in  the  arc  intercepted  by  the  sides  of  an  angle,  is 
taken  as  the  measure  or  value  of  the  angle  itself.  Thus 
the  value  of  the  angle  B  O  D  is  expressed  by  the  number 
of  degrees,  &c.,  contained  in  the  arc  E  F. 

It  is  evident  that  the  absolute  length  of  a  degree  will  differ 
in  different  circles,  being  proportional  to  the  radius  of  the 
circle.  In  other  words,  the  greater  the  radius  of  the  circle, 
the  greater  the  length  of  a  degree  :  whence  it  follows  that  the 
arc  included  between  the  sides  of  an  angle,  whether  these 
sides  be  longer  or  shorter,  will  always  contain  the  same 
number  of  degrees.  Thus,  in  the  figure,  the  arcs  A  C  and 
E  F  contain  each  30  degrees,  or  one-third  of  their  respec- 
tive quadrants.  Hence  we  may  take,  as  the  measure  of  an 
angle,  any  circular  arc  whatever  intercepted  between  the 
sides  of  the  angle,  and  having  its  centre  at  the  vertex  of 
the  angle.  An  angle  of  90°,  or  one  measured  by  a  quad- 
rant, is  called  a  right  angle.  In  the  figure,  D  O  H  is  a 
right  angle.  When  arcs  are  employed  to  express  the 
values  of  angles  in  degrees,  &c.,  they  are  often  called 
angular  distances. 

When  two  planes  intersect  each  other,  they  are  said  to 
form  an  angle  with  each  other,  called  a  diedral  angle :  the 
planes  are  called  the  faces  of  the  angle ;  and  their  line  of 
common  intersection,  the  edge  of  the  angle.  A  diedral  angle 
is  measured  by  the  angle  contained  between  two  lines, — 
one  drawn  in  each  face,  and  both  perpendicular  to  the  com- 
mon intersection  at  the  same  point.  It  is  obvious  that  the 


JO  NEW    METHOD    OF 

angle  formed  by  any  pair  of  lines  thus  drawn  is  equal  to 
the  angle  formed  by  any  other  pair.  An  open  book  may 
serve  to  illustrate  what  is  meant  by  a  diedral  angle.  The 
two  sides  of  the  open  book  are  the  planes,  the  back  of  the 
book  represents  their  intersection,  and  the  angle  of  any 
pair  of  lines  of  the  print  tending  to  meet  at  the  same  point 
of  intersection  on  the  inside  of  the  back  edge  is  the  meas- 
ure of  the  diedral  angle  formed  by  the  planes  of  the  sides. 

Fig.  2.  Fig.  3. 


A  right  diedral  angle.  An  oblique  diedral  angle. 

12.  A  solid  has  length,  breadth,  and  thickness. 

13.  A  sphere  is  a  solid  terminated  by  a  curved  surface, 
all  the  points  of  which  are  equally  distant  from  a  point 
within,  called  the  centre.    A  sphere  may  be  formed  or  gen- 
erated by  the  revolution  of  a  semicircle  about  its  diameter. 

14.  The  radius  of  a  sphere  is  a  straight  line  drawn  from 
the  centre  to  any  part  of  the  surface. 

15.  The  diameter  of  a  sphere  is  a  straight  line  passing 
through  the  centre,  and  terminated  on  both  sides  by  the 
surface.     It  is  equal  to  two  radii. 

16.  Every  section  of  a  sphere  made  by  a  plane  is  a 
circle.     (For  proof,  consult  some  work  on  geometry,  as 
Chauvenet's  Geometry,  p.  245.) 

An  apple,  roughly  speaking,  may  be  considered  as  a 
sphere.  Cut  it  into  two  parts,  and  each  cut  surface  is  a 
circle. 


MOUNTING  AND   OPERATING   GLOBES. 


II 


Fig.  4- 


17.  A  circle  which  divides  a  sphere  into  two  equal  par's, 
called  hemispheres,  is  called  a  great  circle.     Two  great 
circles  of  the  same  sphere  bisect  each  other,  since  their 
line  of  intersection  is  a  diameter  of  the  sphere. 

18.  A  circle  which  divides  a  sphere  into  two  unequal 
parts  is  called  a  small  circle. 

Cut  an  apple  into  two  equal  parts,  and  each  cut  surface 
is  a  great  circle ;  cut  it  into  two  unequal  parts,  and  each 
cut  surface  is  a  small  circle. 

19.  The  poles  of  a  great  circle  of  a  sphere  are  the  ex- 
tremities of  that  diameter 

which  is  perpendicular  to 
the  circle ;  and  these  ex- 
tremities are  also  the 
poles  of  all  small  circles 
parallel  to  the  great  cir- 
cle. Cut  an  apple  into 
slices  across  its  core,  and 
the  stem  and  blossom,  or 
extremities  of  the  core, 
are  the  poles  of  all  the 
slices. 

20.  Angular  Motion. — 
When  a  body  is  continu- 
ally changing  its  direction 

with  respect  to  another  body  which  is  considered  to  be  at 
rest,  it  is  said  to  have  an  angular  motion  around  this  other 
body.  The  simplest  case  of  this  kind,of  motion  is  that  cf 
a  point  proceeding  around  the  centre  of  a  circle  upon  the 
circumference.  The  end  of  the  minute-hand  of  a  watch, 
for  example,  has  an  angular  motion  around  its  pivot,  such 
that  it  describes  a  complete  circle,  or  360°,  every  hour. 

Angular  velocity  is  the  rate  or  degree  of  rapidity  of  angu 
lar  motion,  and  is  measured  by  the  angle  described  in  one 
unit  of  time,  —  second,  minute,  hour,  or  day,  as  the  case 
may  be.     The  angular  distance  passed  over  in  any  time  is 
equal  to  the  angular  velocity  multiplied  by  the  time. 


Great  and  small  circles. 


12  NEW   METHOD    OF 

SECTION     II. 

GEOGRAPHICAL   DEFINITIONS. 

21.  THE  earth,  in  shape,  is  very  nearly  a  sphere,  or  globe. 
More  exactly,  it  has  the  form  of  a  globe  slightly  flattened 
around  the  ends  of  a  certain  diameter,  called,  for  a  reason 
mentioned  hereafter,  the  axis  of  the  earth.     This  figure  is 
known  among  mathematicians  as  the  oblate  spheroid.    The 
amount  of  the  flattening,  in  the  case  of  the  earth,  is  very 
small,  being  only  s^th  part  of  the  length  of  the  axis.     A 
globe  25  feet  in  diameter  would  be  brought  to  the  same  pro- 
portions by  reducing  this  diameter,  in  one  direction,  the 
amount  of  one  inch. 

The  circumference  of  a  great  circle  of  the  earth  is  about 
25,000  miles ;  the  diameter  of  such  a  circle  is  about  8,000 
miles.  The  smallest  or  axial  diameter  of  the  earth  is  26 
miles  less  than  its  greatest  diameter. 

For  the  purpose  of  determining  the  relative  positions 
and  distances  of  points  on  the  earth's  surface,  certain 
imaginary  circles  are  supposed  to  be  drawn  upon  its  sur- 
face. 

22.  A  terrestrial  globe  is  a  hollow  ball  or  sphere,  upon 
whose  surface  is  delineated  a  map  of  the  earth,  having 
drawn  upon  it  the  imaginary  circles  above  referred  to. 

23.  The  axis  of  the  earth  is  the  diameter  about  which 
it  rotates,  like  a  wheel  about  an  axle.     (This  motion  is 
considered  at  length  in  Sect.  IV.) 

24.  The  poles  are  the  extremities  of  the  axis.     The  one 
nearer  to  North  America  is  called  the  north  pole:   the 
other  is  called  the  south  pole. 

25.  The  equator  is  a  great  circle  passing  around  the 
earth  midway  between  the  poles.     It  divides  the  surface 
of  the  earth  into  a  Northern  and  a  Southern  Hemisphere. 


MOUNTING   AND    OPERATING   GLOBES.  13 

26.  Meridians  are  great  circles  passing  through  the 
poles,  and  crossing  the  equator  at  right  angles.  Every 
place  on  the  earth's  surface  may  be  supposed  to  have  a 
meridian  passing  through  it. 


.  —  The  word  "meridian,"  as  generally  used,  means  a  half  of 
the  gre'at  circle  which  it  designates,  rather  than  the  whole. 

27.  Parallels  of  latitude  are  small  circles  parallel  to  the 
equator.     The  farther  a  parallel  is  from  the  equator,  the 
smaller  is  its  diameter.     Every  place  on  the  earth's  surface 
(except  places  on  the  equator)  may  be  supposed  to  have  a 
parallel  passing  through  it. 

28.  The  tropics  are  two  parallels  at  a  distance  of  23^° 
from  the  equator.     The  one  north  of  the  equator  is  called 
the  Tropic  of  Cancer  ;  the  other,  the  Tropic  of  Capricorn. 

29.  The  polar  circles  are  two  parallels  at  a  distance  of 
23  £°  from  the  poles.     The  one  surrounding  the  north  pole 
is  called  the  Arctic  Circle  ;  the  other,  the  Antarctic  Circle. 
(For  the  reason  of  the  position  of  the  tropics  and  polar 
circles,  see  p.  34,  Art.  71.) 

30.  The  tropics  and  polar  circles  divide  the  surface  of 
the  earth  into  five  zones:  namely,  one  torrid,  lying  be- 
tween  the   tropics  ;    two  temperate,  —  a  northern  and  a 
southern,  —  extending   beyond    the    tropics    to   the   polar 
circles  ;  and  two  frigid,  —  a  northern  and  a  southern,  — 
extending  beyond  the  polar  circles  to  the  poles. 

31  Latitude  is  the  angular  distance  of  a  place  from 
the  equator,  measured  on  its  meridian.  Distance  from  the 
equator  towards  the  north  pole  is  called  north  latitude  ; 
from  the  equator  towards  the  south  pole,  south  latitude. 
Places  situated  on  the  equator  have  no  latitude  ;  in  other 
words,  the  latitude  of  such  places  is  zero.  Latitude  near 
the  equator  is  called  low  latitude  ;  near  either  pole,  high 
latitude.  The  highest  latitude  is  90°,  or  the  distance  of 
either  pole  from  the  equator.  Every  point  on  the  same 
parallel  has  the  same  latitude. 


14  NEW    METHOD    OF 

32.  Longitude  is  the  angular  distance   between   two 
meridians,  measured  upon  the  equator  or  a  parallel,  —  one 
of  the  meridians  passing  through  the  place  whose  longi- 
tude is  required ;  and  the  other  being  employed,  by  com- 
mon consent,  to  reckon  from.     In  England,  and  usually  in 
the  United  States,  longitude  is  reckoned  from  the  meridian 
passing  through  Greenwich,  near  London.     Distance  to 
the  right  of  the   meridian  of   Greenwich  is  called   east 
longitude ;   to  the  left,  west  longitude.     Places  situated 
upon  the  meridian  of  Greenwich  have  no  longitude.     The 
meridian  from  which  longitude  is  reckoned  is  often  called 
the  First  Meridian.     Every  point  on  the  same  meridian  (a) 
has  the  same  longitude. 

33.  Any  two  meridians  include  between  them,  whether 
measured  upon  the  equator  or  a  parallel,  an  arc  having  the 
same  number  of  degrees  ;  for  the  planes  of  these  merid- 
ians form  a  diedral  angle,  which  is  measured  by  the  angle  of 
any  pair  of  lines  drawn,  one  in  each  meridian  plane,  from 
a  common  point  of  the  intersection  of  the  plane  (which  is 
the  earth's  axis),  perpendicular  to  the  intersection.     If  the 
pair  of  lines  be  drawn  from  the  middle  of  the  axis,  their 
angle  is  measured  by  the  arc  of  the  equator  included  be- 
tween them ;  if  from  any  other  point  of  the  axis,  by  the 
arc  of  the  corresponding  parallel  included  between  them. 
Therefore  these  arcs  must  contain  the  same  number  of 
degrees.     (See  angle,   including  diedral  angle,  Art.    n.) 
But  the  absolute  length  of   the  intercepted  arcs,  and  of 
each  degree  on  the  arcs,  is  in  proportion  to  the  length  of 
the  radii  of  their  several  circles,  according  to  the  explana- 
tion already  given  in  treating  of  an  angle. 

34.  The  definitions  of  meridians,  parallels  of  latitude, 
&c.,  are   illustrated   in   fig.  5.     Meridians   are   drawn   at 
intervals  of  20°.     Parallels  of  latitude,  including  also  the 


(a)  "  Meridian "  here  really  means  a  semi-meridian  passing  be- 
tween the  poles. 


MOUNTING   AND    OPERATING    GLOBES.  15 

equator,  are  drawn  at  intervals  of  18°  between  the  poles. 
The  tropics  and  polar  circles  are  represented  by  dotted 
lines. 

Fig.  s- 


SECTION     III. 

ASTRONOMICAL   DEFINITIONS. 

35.  THE  imaginary  concave  surface  in  which  a  spectator 
at  first  conceives  all  the  heavenly  bodies  placed  is  a  hemi- 
sphere, in  the  centre  of  the  base  of  which  he  himself  is 
situated.     The  entire  sphere  is  commonly  called,  in  astron- 
omy, the  celestial  sphere.    For  the  purpose  of  defining  the 
positions  of  the  heavenly  bodies  within  this  sphere,  it  is 
found  convenient  to  make  use  of  certain  imaginary  lines, 
and  positions  of  reference,  similar  to  those  which  are  em- 
ployed for  a  like  purpose  on  the  earth's  surface. 

36.  If  the  axis  of  the  earth  were  extended  indefinitely 
in   the  direction  of   the  north  pole,  it  would   pierce   the 
celestial  sphere  in  a  point  called  the  north  celestial  pole ; 


l6  NEW    METHOD    OF 


if  the  axis  of  the  earth  were  extended  indefinitely  in  the 
direction  of  the  south  pole,  it  would  pierce  the  celestial 
sphere  in  a  point  called  the  south  celestial  pole.  These 
points  coincide  with  two  fixed  points  within  the  celestial 
sphere,  around  which  the  stars  that  do  not  rise  and  set 
appear  to  move  in  circles. 

37.  If  the  plane  of  the  terrestrial  equator  were  extended 
indefinitely,  it  would  intersect  the  celestial  sphere  in  a 
great  circle,  called  the  celestial  equator. 

38.  If  the  plane  of  the  terrestrial  meridian  of  a  place 
were  extended  indefinitely,  it  would  intersect  the  celestial 
sphere  in  a  great  circle,  called  the  celestial  meridian  of 
the  place. 

39.  Circles  or  Parallels  of  declination  are  small  circles 
of  the  celestial  sphere  parallel  to  the  equator. 

40.  Hour  circles  are  great  circles  of  the  celestial  sphere 
passing  through  the  poles,  and  intersecting  the  equator  at 
right  angles. 

41.  The  base  of  the  celestial  hemisphere  which  is  visi- 
ble at  any  place  is  called  the  plane  of  the  horizon.     The 
great  circle  in  which  this  plane   intersects  the  celestial 
sphere   is   called   the   horizon.      Astronomers   sometimes 
distinguish  between  two  horizons,  called  respectively  the 
sensible   and   the  rational    horizon.     This   distinction   is 
rather   between   the   planes  of    these  horizons  than  the 
horizons  themselves.     The  plane  of  the  sensible  horizon 
at  any  place  is  an  imaginary  plane  passing  through  the 
place,  perpendicular  to  the  earth's  radius  at  that  place. 
The  plane  of  the  rational  horizon  of  the  place  is  a  plane 
parallel  to  that  of  the  sensible  horizon,  and  passing  through 
the  earth's  centre.    These  two  planes  apparently  meet  in  the 
same  great  circle,  at  the  distance  of  the  celestial  sphere. 

42.  The  zenith  is  the  point  within  the  celestial  sphere 
which  is  over  the  head  of   a  spectator  upon  the  earth's 
surface.     The  nadir  is  the  point  which  is  under  his  feet, 
or  diametrically  opposite  his  zenith. 


MOUNTING   AND    OPERATING    GLOBES.  I/ 

43.  Vertical  circles  are  great  circles  passing  through  the 
zenith  and  nadir,  and  intersecting  the  horizon  at  right  angles. 

That  vertical  circle  which  intersects   the  celestial   me- 
ridian at  right  angles  is  called  the  prime  vertical. 

44.  The  four  points  where  the  meridian  and  prime  ver- 
tical intersect   the   horizon  of   any  place   are  called  the 
cardinal  points  of  the  horizon ;  those  of  the  meridian,  the 
north  and  south  points  ;  those  of  the  prime  vertical,  the  east 
and  west  points.    These  latter  points  are  evidently  the  poles 
of  the  meridian  ;  and,  since  the  celestial  equator  is  a  great 
circle  perpendicular  to  the  meridian,  it  must  pass  through 
them. 

45.  The  altitude  of  a  celestial  body  is  its  angular  height 
above  the  horizon,  measured  on  a  vertical  circle.     When 
this  distance  is  measured  on  the  celestial  meridian  of  any 
given  place,  it  is  called  meridian  altitude. 

46.  The  zenith  distance  of  a  celestial  body  is  its  angu- 
lar distance  from  the  zenith.     It  is  the  complement  of  the 
altitude. 

47.  The  azimuth  of  a  celestial  body  is  its  angular  dis- 
tance, measured  on  the  horizon,  between  the  north  or  south 
points  (according  to  the  hemisphere  in  which  the  observa- 
tion is  made)  and  the  point  in  which  a  vertical  circle  pass- 
ing through  the  body  intersects  the  horizon. 

48.  The  amplitude  of    a  celestial  body  is  its  angular 
distance  from  the  east  point  when  it  rises,  or  from  the 
west  point  when  it  sets. 

49.  The  declination  of  a  celestial  body  is  its  angulai 
distance  from  the  equator,  either  north  or  south.     Declina- 
tion corresponds  to  terrestrial  latitude. 

50.  The  right  ascension  of  a  celestial  body  is  the  arc  of 
the  celestial  equator  included  between  two  hour-circles,  — 
one  passing  through  the  body;  and  the  other  through  a 
point  on  the  celestial  equator,  called  the  vernal  equinox. 
Right  ascension  corresponds  to  longitude  on  the  earth  ;  but, 
unlike  longitude,  it  is  always  reckoned  towards  the  east. 


18 


NEW    METHOD    OF 


51.  At  the  equator  the  celestial  poles  are  in  the  horizon  ; 
and  hence  the  celestial  equator,  and  parallels  of  declination, 
are  all  perpendicular  to  the  horizon.  This  position  of  the 
sphere  is  called  a  right  sphere. 


Fig.  6. 


Fig.  7. 


Right  sphere. 


Parallel  sphere. 


Fig.  8. 


52.  At  the  terrestrial  poles,  the  celestial  poles  appear 
in  the  zenith,  and  the  celestial  equator  coincides  with  the 

horizon.  The  parallels  of  dec- 
lination are  parallel  to  the  ho- 
rizon. This  position  of  the 
sphere  is  called  a  parallel 
sphere. 

53.  At   all    places,   except 
the    equator   and    poles,    the 
celestial  equator,   and  paral- 
lels of  declination,  are  oblique 
to  the  horizon.    This  position 
of  the  sphere  is  called  an  ob- 
lique sphere. 

54.  The  apparent  path  of 
the  sun  traced  on  the  surface 

of  the  celestial  sphere,  among  the  fixed  stars,  is  a  great 


Oblique  sphere. 


MOUNTING   AND    OPERATING   GLOBES.  19 

circle,  which  he  moves  over  in  a  direction  from  west  to 
east.  This  circle  is  called  the  ecliptic,  because  eclipses 
take  place  when  the  moon,  at  the  new  and  full,  is  in  or 
near  this  circle.  The  ecliptic  is  inclined  to  the  celestial 
equator  at  an  angle  of  about  23^°. 

In  order  that  we  may  define  at  any  time  the  position  of 
the  sun  in  the  ecliptic,  this  circle  is  supposed  to  be  divided 
into  12  equal  parts,  called  signs,  each  containing  30°. 
These  signs  are  distinguished  into  Spring,  Summer,  Au- 
tumn, and  Winter  signs.  These,  with  their  symbols,  are 
as  follows :  — 


Spring  Signs. 

T  Aries, 
s  Taurus, 
n  Gemini. 

Summer  Signs. 

o  Cancer. 
ft  Leo. 
mi  Virgo. 

Autumn  Signs. 

•*  Libra. 
"I  Scorpio. 
*  Sagittarius. 

Winter  Signs. 

«  Capricornus. 
~  Aquarius. 
*  Pisces. 

The  sun  enters  these  signs  upon  dates  varying  from  the 
to  the  23d  of  the  several  months.  For  the  sake  of 
convenience,  the  signs  are  often  considered  as  beginning  on 
the  2ist  of  each  month. 

The  cardinal  signs  of  the  ecliptic  are  Aries,  Cancer, 
Libra,  and  Capricornus.  There  are  four  cardinal  points 
on  the  ecliptic,  called  the  equinoctial  and  solstitial  points, 
which  at  present  have  the  following  positions  for  the  north- 
ern hemisphere  :  Vernal  equinox,  the  first  degree,  or,  as  it  is 
usually  called,  the  first  point,  of  Aries  ;  autumnal  equinox, 
the  first  point  of  Libra ;  summer  solstice,  the  first  point  of 
Cancer;  winter  solstice,  the  first  point  of  Capricornus. 
These  points  mark  important  positions  of  the  sun  within 
the  ecliptic,  as  referred  to  the  equator :  thus  the  solstitial 
points  are  the  points  in  which  the  sun  attains  its  farthest 
positions  north  or  south  of  the  equator;  and  the  equinoctial 
points  are  the  points  in  which  the  sun  crosses  the  equator, 
or  where  the  equator  and  the  ecliptic  intersect  each  other. 

55.  Since  the  circles  of  the  celestial  sphere  coincide  in 


2O  NEW   METHOD    OF 

position  with  those  defined  upon  the  earth's  surface  (except- 
ing the  ecliptic),  they  may  be  represented  by  the  latter  as 
drawn  upon  a  terrestrial  globe.  The  ecliptic,  to  which 
no  terrestrial  circle  corresponds,  may  therefore  have  its 
relative  position  within  the  heavens  represented  either 
upon  a  celestial  or  a  terrestrial  globe. 


SECTION    IV. 

PHENOMENA   CAUSED   BY  THE   MOTIONS   OF   THE   EARTH. 

56.  EVERY  morning,  the  sun  makes  his  appearance  upon 
the  eastern  horizon,  ascends  above  it  until  noonday,  then 
descends,  disappearing  at  last  beneath  the  western  hori- 
zon.    At  night,  the  moon  is  often  seen  pursuing  a  similar 
track  across  the  heavens;   and  the  watcher  of   the  stars 
sees  them  one  by  one  rise,  describe  a  course  along  the 
sky,  and  set,  until  the  sun  again  returns  to  eclipse  them 
with  his  superior   light,  and  to  repeat   his  daily  journey 
towards  the  west.     Again :  if,  at  the  same  hour  for  sev- 
eral successive  nights,  the  position  of  a  star,  or  group  of 
stars,  be  noted,  it  will  appear  (a)  to  be  moving  westward 
at  the  rate  of  about  i°  during  the  twenty -four  hours,  this 
constant  change  of  position  amounting  to  an  entire  circuit 
during  a  year. 

57.  We  thus  recognize  two  distinct  movements  of  the 
heavenly  bodies  ;  namely,  a  daily  motion  of  revolution,  and 
a  motion  of  revolution  which  is  completed  in  just  a  year. 
The  question  now  arises,  Are  these  two  movements  real, 
or  only  apparent  ?  that  is,  do  they  belong  to  the  heavenly 
bodies  themselves,  or  has   the  earth  a  system  of  move- 


(a)  The  fact  being  that  the  stars  are  at  rest  while  the  sun  really 
moves  eastward. 


MOUNTING   AND   OPERATING   GLOBES.  21 

ments  to  which  they  may  be  referred,  as  objects  upon  a 
roadside  change  their  positions  with  reference  to  a  passing 
carriage  ? 

58.  It  may  appear,  at  first  thought,  strange  to  associ- 
ate  ideas   of    motion   with    an    earth   which   has    always 
seemed  so  stable  to  its  inhabitants  ;  but  experience  teaches 
us  that  it  is  not  at  all  times  an  easy  matter  to  distinguish 
between  the  two  states  of  motion  and  of  rest.     When,  for 
example,  we  travel  by  night,  deprived  of  the  view  of  all 
objects  outside  of  the  vehicle  in  which  we  are  seated,  we 
are  conscious  of  no  sense  of  progress,  notwithstanding  that 
we  feel  the  tremor  with  which  the  vehicle  is  carried  over 
the  inequalities  of    the   road.      The   traveller  who  looks 
from  a  car-window,  as  he  journeys  swiftly  along,  receives 
a  very  strong  impression  that  the  objects  of  the  landscape 
are  making  as  swift  a  transit  in  an  opposite  direction. 
The  aeronaut  testifies,  that,  as  he  rises  or  descends  in  his 
car,  it  is  the  earth  which  seems  to  move  from  him,  or  to 
return  to  him.     A  ship  swings  round  in  a  land-locked  bay, 
and  one  looking  out  of  the  cabin-window  fancies  the  whole 
shore  to  pass  in  the  direction  of  a  circle  around  him.    And 
so  we  might  continue  to  enumerate  instances  tending  to 
prove  that  the  senses  alone,  unassisted  by  experience  or 
judgment,  are  often  incapable  of  ascribing  a  change  of 
place  to  its  proper  source. 

59.  It  is  evident,  then,  that,  so  far  as  our  vision  is  con- 
cerned (and  this  is  the  only  sense  by  which  we  are  made 
aware  of  the  movements  of  the  heavenly  bodies),  these 
movements  may  belong  either  to  the  heavenly  bodies  them- 
selves, or,  with  just  as  good  a  show  of  reason,  to  the  earth. 
Now,  which  is  the  more  probable  supposition,  considering 
at  first  only  the  daily  motion  ?     Do  thousands,  or  rather 
millions,  of  celestial  bodies  at  vast  distances,  and  many 
of  them  of  immense  magnitudes,  perform  a  diurnal  revolu- 
tion about  the  earth  in  twenty-four  hours?    or   does   the 
earth   rotate   about   an    axis   in   this  time?      The    latter 


22  NEW    METHOD    OF 

hypothesis  would  ascribe  the  effect  to  the  simplest  cause, 
and  is  unquestionably  the  more  reasonable  one. 

60.  But  there  are  experiments  and  reasons,  which,  to  the 
minds  of  those  who  understand  them,  are  as  convincing 
proofs  of  the  earth's  rotation  as  though  they,  from  some  dis- 
tant stand-point,  actually  saw  this  rotation  going  on.  In  the 
first  place,  it  has  been  found  that  the  weight  of  a  body  in- 
creases when  carried  from  the  equator  towards  the  poles. 
Now,  the  generation  of  a  centrifugal  force  by  means  of  rota- 
tion —  this  force  being  the  greatest  where  the  earth's  surface 
has  the  swiftest  motion,  or  at  the  equator  —  would  ac- 
count for  such  a  variation  in  weight ;  for  such  a  force,  being 
directed  away  from  the  earth's  centre,  diminishes,  accord- 
ing to  its  amount,  the  weight  of  a  body,  which  is  a  force 
directed  towards  the  centre  of  the  earth. 

Experiments  of  another  character  have  demonstrated  that 
the  very  shape  of  the  earth  is  a  strong  proof  of  its  rotation. 
This  shape,  as  we  know,  is  that  of  a  sphere  flattened  at  the 
poles,  and  bulging  at  the  equator.  Now,  it  is  found  that 
when  a  rotation  equal  to  that  of  the  earth  in  amount,  or  per- 
formed once  in  twenty-four  hours,  is  given  to  a  fluid  or  semi- 
fluid mass,  this  soon  assumes  the  same  spheroidal  shape 
which  the  earth  is  known  to  have.  Thus  a  mass  of  oil, 
placed  within  a  transparent  liquid  of  the  same  density,  re- 
tains the  form  of  a  perfect  sphere  while  at  rest,  but  changes 
this  form  for  that  of  a  spheroid  the  moment  a  slow  motion 
of  rotation  is  given  to  it  by  means  of  a  piece  of  wire  forced 
through  it.  The  theory  that  the  earth  was  once  intensely 
heated,  and  therefore  fluid,  is  rendered  highly  probable  by 
geological  and  cosmical  considerations,  and  by  the  still 
heated  condition  of  the  interior  of  the  earth. 

There  are,  also,  experimental  proofs  of  the  earth's  rota- 
tion, of  the  most  direct  and  conclusive  character,  such  as 
Foucault's  famous  pendulum  experiment ;  but,  for  an  ex- 
planation of  these,  the  learner  must  consult  other  and 
larger  works. 


MOUNTING   AND    OPERATING    GLOBES.  23 

61.  The   daily   or  nightly   movement  of  the   heavenly 
bodies  is  much  more  evident  to  the  eye  than  the   much 
slower  annual  motions  of  certain  of  them,  like  the  sun  and 
moon.     The  latter,  however,  may  be  readily  observed  by 
comparing  the  positions  of  these  bodies  with  respect  to  the 
fixed  stars  for  a  few  consecutive  nights.     So  far  as  the  sun 
is  concerned  (which  is  the  only  body  whose  annual  move- 
ment we  consider),  its  annual  motion  consists  in  the  descrip- 
tion of  a  great  circle  among  the  fixed  stars  already  defined 
as  the  ecliptic.     The  apparent  annual  motion  of  the  sun 
within  this  circle  may  be  explained  by  supposing  either 
that  the  sun  moves  round  the  earth,  or  the  earth  round  the 
sun,  just  as  we  had  a  similar  choice  of  explanation  when 
considering  the  apparent  daily  motion  of  the  same  body. 
That  the  latter  is  the  case  is  rendered  probable  in   the 
highest  degree  from  certain  phenomena,  as  the  aberration 
of  light,  and  certain  well-known  physical  laws,  an  explana- 
tion of  which  belongs  rather  to  a  work  upon  astronomy 
than  to  the  present  brief  treatise. 

62.  The  annual  motion  of  the  earth  round  the  sun  is  per- 
formed in  such  a  manner,  that  at  all  times  the  earth  and  the 
sun  lie  very  nearly,  or,  as  we  shall  suppose,  exactly,  in  one 
and  the  same  plane,  called  the  plane  of  the  ecliptic.     The 
intersection  of  this  plane  with  Fig 

the  surface  of  the  heavens  is 
the  ecliptic,  already  otherwise 
defined  in  Art.  54.  In  this 
plane  the  orbit  of  the  earth  is  a 
certain  curve,  called  in  geome- 
try an  ellipse;  a  figure  differing 
from  a  circle  by  being  more  or 
less  oval  in  form,  and  also  in 
having  two  points  within,  to 
which  its  circumference  is  referred,  instead  of  one  central 
point,  as  in  the  circle.  These  two  points  are  called  the 
foci  of  the  ellipse,  and  in  one  of  these  foci  the  sun  is  situ- 
ated. In  the  figure,  A  and  B  represent  these  points. 


24  NEW   METHOD    OF 

The  distance  of  A  or  B  from  the  nearest  point  in  the 
ellipse  of  Fig.  9  is  much  less  than  their  distance  apart; 
but  in  the  case  of  the  orbit  of  the  earth  it  is  quite  the 
contrary,  the  foci  there  being  very  much  nearer  to  each 
other  than  they  are  distant  from  their  ellipse.  This  is 
because  the  orbit  of  the  earth  approaches  much  more 
nearly  the  form  of  a  circle  than  the  ellipse  of  Fig.  9. 

Now,  since  the  sun  does  not  occupy  the  centre  of  the 
earth's  orbit,  but  one  of  the  foci,  it  follows  that  the  earth  is 
nearer  to  it  at  one  time  than  another.  When  these  two 
bodies  are  nearest  together,  we  say  the  earth  is  in  perihelion; 
when  they  are  farthest  apart,  we  say  it  is  in  aphelion.  These 
two  positions  are  represented  in  Fig.  12,  page  31.  The  earth 
advances  around  the  sun  at  its  slowest  rate  when  it  is  in 
aphelion,  and  at  its  fastest  rate  when  it  is  in  perihelion.  It 
is  in  perihelion  upon  January  i,  and  in  aphelion  upon  July  8. 
The  distance  of  the  earth  from  the  sun,  when  in  perihelion, 
is  about  90,000,000  miles ;  when  in  aphelion,  93,000,000 ; 
the  former  distance  being  f  §ths  of  the  latter. 

The  orbit  of  the  earth  is  more  than  half  a  billion  miles  in 
extent.    The  earth  moves  in  this 
Fig-  I0>  orbit  at  the  rate  of  68,000  miles 

an  hour,  or  with   a   velocity  of 
about  19  miles  a  second. 

63.  The  earth  moves  in  its 
orbit  from  west  to  east.  A  mo- 
tion of  revolution  from  west  to 
east  may  be  defined  as  motion 
in  a  direction  contrary  to  that  in 
which  the  hands  of  a  clock  move, 
the  spectator  being  supposed  to 
be  situated  at  the  north  pole, 

with  his  face  towards  the  sun.  The  direction  in  which 
the  sun  appears  to  move  along  the  ecliptic  is  the  same 
as  that  of  the  real  motion  of  the  earth  in  its  orbit,  or 
towards  the  east.  Fig.  10  illustrates  the  relative  direction 


MOUNTING   AND    OPERATING    GLOBES.  2$ 

of  the  earth  and  sun  in  their  respective  yearly  courses, 
real  and  apparent.  When  the  earth  is  at  A,  the  sun  will 
be  seen  in  the  direction  of  the  line  A  D,  or  as  if  projected 
on  the  surface  of  the  sky  among  the  stars  at  D.  When  the 
earth  has  reached  the  position  B,  the  sun  will  be  seen  pro- 
jected upon  E ;  while,  with  the  earth  in  the  position  C,  the 
sun  will  be  seen  projected  upon  F.  The  sun  has  therefore 
seemed  to  move  from  D  to  F,  while  the  earth  has  advanced 
from  A  to  C  ;  and  thus  the  apparent  movement  would  con- 
tinue around  the  entire  circle  in  a  direction  towards  the 
east,  or  the  same  as  that  of  the  earth  in  its  orbit.  This 
sameness  of  direction  is,  on  the  contrary,  not  met  with  in 
the  real  and  the  apparent  diurnal  motions,  the  directions 
of  these  being  opposed  to  each  other :  the  sun  moves  from 
east  to  west  across  the  sky,  while  the  earth  rotates  upon  its 
axis  from  west  to  east. 

64.  The  axis  of  the  earth  is  not  perpendicular  to  the 
plane  of  its  orbit,  or  plane  of  the  ecliptic,  but  is  inclined 
towards  it  23^°  ;  that  is,  the  axis  makes  an  angle  of  23^° 
with  a  line  drawn  perpendicular  to  the  plane  of  the  earth's 
orbit,  and  therefore  an  angle  of  90-23 J°  or  66^°  with  the 
plane  of  the  orbit  itself. 

It  follows,  from  this  inclination  of  the  earth's  axis  to  the 
plane  of  its  orbit,  or  plane  of  the  ecliptic,  that  this  plane 
has  also  an  inclination  of  23^°  to  the  plane  of  the  equator, 
as  stated  on  page  19.  This  may  be  illustrated  by  sup- 
posing the  earth  to  move  around  the  sun  half  immersed 
within  the  surface  of  an  ocean,  and  having  its  axis  in 
an  upright  position,  when  the  equator  would  everywhere 
rest  upon  the  ocean,  or  the  planes  of  the  ecliptic  and  of 
the  equator  would  coincide.  Now,  incline  the  axis  one 
or  more  degrees,  and  straightway  the  equator,  which  is  a 
part  of  the  same  sphere,  falls  below  the  ocean  the  same 
number  of  degrees  along  half  of  its  extent,  and  rises  above 
it  the  same  number  of  degrees  along  the  other  half  ;  or  the 
planes  of  the  ecliptic  and  of  the  equator  now  form  an 


26  NEW    METHOD    OF 

angle  with  each  other,  just  equal  to  that  formed  by  the 
earth's  axis  with  a  perpendicular  to  the  plane  of  the 
ecliptic. 

65.  During  the  annual  journey  of  the  earth  around  the 
sun,  its  axis  constantly  preserves  the  same  direction  in 
space ;  or,  to  express  the  same  fact  in  other  words,  the 
earth 's  axis  always  remains  parallel  to  itself.  An  illustra- 
tion of  this  kind  of  motion  is  presented  in  the  figure,  where 

a  pencil  is  held  in  one  di- 
rection while  it  describes 
a  circle.  The  four  posi- 
tions of  the  pencil  are 
all  parallel  to  each  other, 
and  to  any  number  of 
positions  which  may  be 
represented,  so  long  as 
the  direction  of  the  pen- 
cil remains  the  same. 
In  the  same  way,  the  axis 
of  the  earth,  along  with 
the  earth,  moves  round  the  sun,  preserving  a  constant  di- 
rection all  the  time. 

It  may  be  said  that  the  pencil,  in  its  several  posi- 
tions, is  directed,  above  and  below,  towards  different 
points  in  the  space  beyond ;  whereas  we  know,  by  obser- 
vation, that  the  earth's  axis  always  appears  to  be  directed 
towards  the  same  points  in  the  heavens.  The  reason  of 
this  lies  in  the  fact  that  the  distances  from  the  earth  of  the 
only  visible  fixed  points  in  the  sky  (namely,  the  fixed  stars) 
are  so  enormous,  that,  by  comparison,  even  the  diameter  of 
the  earth's  orbit  (about  180,000,000  miles)  shrinks  into  all 
but  a  mere  point.  The  diameter  of  the  earth's  orbit,  when 
stated  in  miles,  is  a  length  so  great,  that  the  mind  fails  to 
form  an  adequate  conception  of  it.  What,  then,  can  be 
said  of  the  distance  of  even  the  nearest  fixed  star,  except 
that  it  is  known  to  be  more  than  200,000  times  as  great, 


MOUNTING   AND    OPERATING    GLOBES.  2/ 

and  that  light,  travelling  190,000  miles  a  second,  requires 
3^  years  to  pass  from  the  star  to  the  earth?  Viewed  from 
a  fixed  star,  therefore,  not  only  the  earth,  but  the  earth's 
orbit,  becomes  a  mere  point  in  space ;  and,  on  the  other 
hand,  it  follows,  that,  as  seen  from  any  part  of  the  earth's 
orbit,  a  fixed  star  must  always  appear  to  have  the  same 
position  in  the  sky. 

66.  The  alternation  of  day  and  night  is  caused  by  the 
rotation  of  the  earth  on  its  axis  once  every  24  hours. 
The  earth,  therefore,  at  every  instant  is  divided  into  two 
shifting  hemispheres,  one  illuminated  by  the  sun,  and 
the  other  not  illuminated.  If  the  sun  were  a  small 
shining  body  placed  very  near  the  earth,  it  would  illu- 
minate only  a  small  portion  of  its  surface.  This  may  be 
readily  seen  by  holding  a  lighted  match  or  taper  near  a 
sphere  or  globe  in  a  dark  room.  Now,  the  larger  the  lumi- 
nous body,  the  greater  the  extent  of  surface  illuminated 
by  it ;  also,  the  more  distant  the  luminous  body  (the  light 
being  supposed  to  be  sufficiently  intense),  the  greater  the 
amount  of  surface  thus  illuminated.  In  the  case  of  the 
sun,  its  size  and  distance,  and  the  intensity  of  its  light,  are 
so  great,  that  it  illuminates  almost  precisely  one-half  of  the 
earth's  surface  at  ajl  times.  In  other  words,  at  every  in- 
stant the  sun  is  above  the  horizon  at  all  places  on  one-half 
of  the  earth's  surface,  and  below  it  at  all  places  on  the 
other  half.  The  line  around  the  earth  separating  the  sur- 
faces of  the  illuminated  and  unilluminated  hemispheres 
must  be  a  great  circle.  This  great  circle  is  often  called 
the  circle  of  illumination,  or,  when  represented  on  a  terres- 
trial globe,  the  day-circle,  —  a  name  by  which  we  shall  call 
it  hereafter. 

In  order  to  illustrate  the  alternation  of  day  and  night, 
we  may  again  imagine  the  earth  floating  on  an  ocean,  with 
exactly  one  hemisphere  below  the  surface  of  the  water. 
Then  the  hemisphere  which  is  above  the  surface  may  fitly 
represent  the  hemisphere  which  is  illuminated  by  the  sun's 


28  NEW    METHOD    OF 

rays,  while  the  immersed  hemisphere  answers  to  that  half 
of  the  earth's  surface  over  which  night  prevails.  To  make 
the  conditions  as  simple  as  possible,  suppose  further,  that 
the  poles  of  the  earth,  with  the  entire  axis  of  course,  lie 
exactly  on  the  surface  of  the  water.  Then  it  is  plain  that 
exactly  one-half  of  every  parallel  (equator  included)  must 
be  above  the  surface  of  the  water,  —  that  is,  in  light ;  and 
that  the  other  half  must  be  below  this  surface,  —  that  is, 
in  darkness :  for  the  centre  of  every  parallel  is  situated  in 
the  axis,  and  therefore  in  the  surface  of  the  water.  Now, 
let  the  earth  be  supposed  to  rotate  uniformly  on  its  axis  in 
24  hours,  and  it  is  evident  that  every  place  on  every  paral- 
lel must  be  just  12  hours  above  the  surface  of  the  water, 
or  in  light,  and  12  hours  below,  or  in  darkness  ;  or,  in  other 
words,  under  these  conditions  there  would  be  a  day  and 
a  night  of  12  hours  each  over  the  entire  earth. 

67.  Such  a  rotation  of  the  earth  as  we  have  described 
must  produce,  every  24  hours,  two  equal  periods,  — one  of 
light,  or  day ;  the  other  of  darkness,  or  night.  Now,  as  a 
matter  of  fact,  except  at  the  equator,  this  is  true  only  twice 
a  year ;  viz.,  on  March  21  and  September  21.  At  all  other 
times,  as  we  well  know,  these  periods  are  not  only  different 
in  length,  but,  moreover,  they  undergo  variations  in  length, 
in  the  course  of  the  year.  And  still  further :  the  length  of 
the  day  (or  the  night)  is  different  on  different  parallels  at 
the  same  time  of  the  year.  Let  us  consider  the  chief  facts 
respecting  these  differences  in  something  like  regular  order. 
In  the  first  place,  in  this  part  of  the  world,  we  have  the  long- 
est day  in  June,  and  the  shortest  in  December  ;  while,  as 
to  the  nights,  it  is  just  the  other  way,  —  the  shortest  occur- 
ring in  June,  and  the  longest  in  December.  We  might 
know  that  the  longest  day  must  correspond  to  the  shortest 
night  from  the  fact  that  the  sum  of  the  two  must  be  al- 
ways the  same ;  viz.,  24  hours,  or  the  time  of  one  revo- 
lution of  the  earth  on  its  axis.  Moreover,  as  we  all  know, 
the  length  of  the  day  gradually  diminishes  from  about  the 


MOUNTING   AND    OPERATING   GLOBES.  2Q 

2ist  of  June  to  the  2ist  of  December,  when  it  reaches  its 
shortest  limit:  it  then  begins  to  increase,  at  first  slowly, 
afterwards  more  rapidly  through  the  months  of  late  winter 
and  of  spring,  until,  when  the  2ist  of  June  again  comes 
round,  it  again  reaches  its  greatest  length.  During  the 
succeeding  year,  it  goes  through  the  same  changes  in  the 
same  order.  On  the  other  hand,  the  changes  in  the  length 
of  night,  of  course,  are  exactly  the  reverse  of  those  of  day. 

If  we  lived  in  some  other  part  of  the  world,  —  say,  at  a 
place  nearer  the  north  pole,  as  the  southern  part  of  Green- 
land, —  we  should  find  that  similar  changes  in  the  lengths 
of  day  and  night  occurred  during  the  year,  but  with  this 
important  difference,  —  that  they  were  much  greater  in 
amount.  For  instance  :  June  2 1  is  the  longest  day  in  the 
year  ;  but  instead  of  being  about  15  hours'  duration,  as 
at  Boston,  it  lasts  at  least  20  hours,  followed  by  a  short 
night  of  4  hours,  or  rather  a  period  of  4  hours  of  twilight ; 
since  the  sun  remains  so  near  the  northern  horizon,  that 
the  light  of  day  is  but  partly  removed  even  at  midnight. 
On  the  other  hand,  if  we  take  a  position  near  the  equator, 
we  find,  that,  while  the  same  changes  occur,  they  are  less 
in  amount  than  with  us.  Finally,  on  the  equator  itself, 
there  is  no  change  in  the  duration  of  day  and  night,  each 
being  12  hours  long  throughout  the  year.  In  the  southern 
hemisphere,  it  is  found  that  changes  take  place  in  the  exact 
inverse  order  to  those  which  occur  in  the  northern  hemi- 
sphere. For  example,  when  the  day  is  16  hours  long 
at  a  place  north  of  the  equator,  there  is  a  night  of  16 
hours  at  any  place  which  is  just  as  far  south  of  the 
equator.  It  appears,  in  short,  that  these  changes  in  the  du- 
ration of  day  and  night  are  in  some  way  dependent  upon 
the  latitude  of  the  place,  being  always  greater  in  high  lati- 
tudes than  in  low  ones. 

68.  Still  further:  it  is  a  matter  of  common  observa- 
tion, that  the  sun  rises  and  sets  at  very  different  points 
of  the  horizon  during  the  year ;  and  also,  that,  at  the  hour 


3O  NEW   METHOD   OF 

of  noon,  it  is  much  nearer  the  zenith  at  certain  seasons  of 
the  year  than  at  others.  In  summer,  when  the  days  are 
long,  it  rises  far  to  the  north  of  the  east  point  of  the  hori- 
zon ;  mounts  up  into  the  heavens,  until  at  noon  it  is  nearly 
overhead ;  then  sets  at  a  point  as  much  to  the  north  of  west 
as  the  rising  point  was  north  of  east.  In  mid-winter,  on 
the  other  hand,  when  the  days  are  short,  the  sun's  brief 
course  through  the  sky  lies  low  down  towards  the  southern 
horizon.  It  rises  in  the  south-east,  and  sets  in  the  south- 
west ;  and,  even  at  noon,  is  not  far  above  the  southern  point 
of  the  horizon.  Thus  it  would  seem  that  the  noonday  height 
of  the  sun  is  in  some  way  intimately  related  to  the  length  of 
the  day;  and  we  are  very  naturally  led  to  suspect  that 
the  changes  in  both  are  due  to  the  action  of  a  common 
cause.  This  cause  is  to  be  found  by  a  study  of  the  an- 
nual motion  of  the  earth  round  the  sun  in  the  plane  of 
the  ecliptic,  with  its  axis  preserving  a  constant  direction  in 
space  at  all  times,  as  explained  in  Art.  65.  We  might 
easily  study  this  motion  by  the  aid  of  the  mental  image  of 
the  earth  moving  in  an  ocean,  which  we  have  already  em- 
ployed in  Arts.  64  and  66. 

69.  But  we  shall  obtain  a  more  comprehensive  view  of 
the  subject  by  examining  directly  the  various  positions  which 
the  earth  has  relatively  to  the  sun  during  the  year,  as  they 
are  represented  in  Fig.  12.  In  this  figure,  twelve  positions 
of  the  earth  in  its  orbit  are  shown ;  and  in  all  these  posi- 
tions the  earth's  axis  is  represented  as  constant  in  direc- 
tion, and  making  an  angle  of  66£°  with  the  plane  of  the 
ecliptic. 

Twice  in  the  year,  —  viz.,  at  the  positions  marked  ver- 
nal and  autumnal  equinox,  —  the  sun  shines  from  pole 
to  pole,  and  is  vertical  at  the  equator.  This  is  upon 
March  21  and  September  21.  In  these  two  positions,  in 
fact,  the  earth  has  exactly  the  same  situation  with  re- 
gard to  the  sun  that  has  been  more  fully  explained  in 
Art.  66.  Upon  these  two  dates,  therefore,  day  and  night 


MOUNTING   AND    OPERATING    GLOBES.  3! 

are  everywhere  12  hours  in  length,  or  the  parallels  of  the 
earth's  surface  are  divided  into  two  equal  parts  by  the  day- 


circle.  The  equator  must  be  thus  divided  by  the  day- 
circle  at  all  times,  since  both  are  great  circles,  and  there- 
fore must  always  bisect  each  other. 


32  NEW    METHOD    OF 

March  21  and  September  21,  from  the  equality  of  day 
and  night  at  these  times  over  all  the  earth's  surface  (unless 
we  except  the  poles),  are  called  the  equinoxes  («),  —  the 
former  the  vernal,  and  the  latter  the  autumnal.  7"ne  names 
of  these  two  dates  are  reversed  in  the  southern  hemi- 
sphere ;  March  2 1  being  the  autumnal,  and  September  2 1 
the  vernal. 

Upon  June  21  (see  Fig.  12),  the  sun  shines  over  the  en- 
tire north  frigid  zone  (which  is  included  within  the  small- 
est parallel  represented),  and  leaves  the  entire  south  frigid 
zone  in  darkness.  Upon  December  21,  the  sun  shines 
over  the  entire  south  frigid  zone,  and  leaves  the  entire 
north  frigid  zone  in  darkness.  These  two  dates  are  called 
the  solstices  (£),  since  at  these  times  the  sun  is  stationary 
as  regards  his  approach  to  the  poles.  June  2 1  is  the  sum- 
mer solstice  of  the  northern  hemisphere,  and  the  winter 
solstice  of  the  southern  hemisphere.  December  21  is  the 
winter  solstice  of  the  northern  hemisphere,  and  the  summer 
solstice  of  the  southern  hemisphere. 

These  are,  then,  four  cardinal  positions  of  the  earth,  as 
regards  the  sun  ;  namely,  the  positions  of  the  equinoxes 
and  the  solstices,  these  positions  being  attained  about  the 
dates  of  March  21,  September  21,  June  21,  and  December 
21.  The  four  seasons  of  the  year,  astronomically  consid- 
ered, begin  with  these  dates. 

70.  Let  us  now,  with  the  aid  of  Fig.  12,  see  how  the 
change  of  day  and  night  goes  on  between  these  several 
dates. 


(a)  From  the  two  Latin  words  czqwis  and  nox,  meaning  equal  and 
night. 

(V)  From  the  two  Latin  words  sol  and  stare,  meaning  the  sun  and 
to  stand. 

The  equinoxes  and  solstices,  properly  speaking,  are  times  or  dates 
as  above  defined;  although  these  terms  are  often  applied  to  the 
points  of  the  ecliptic  corresponding  to  the  dates,  and,  indeed,  have 
been  so  defined  on  page  19. 


MOUNTING   AND    OPERATING   GLOBES.  33 

From  June  21  to  September  21,  in  consequence  of  the 
motion  of  the  earth  in  its  orbit,  the  north  frigid  zone 
gradually  has  one-half  of  its  area  removed  from  the  sun's 
light,  while  exactly  the  converse  holds  true  of  the  south 
frigid  zone.  Day  is  therefore  decreasing  over  the  northern 
hemisphere,  and  increasing  over  the  southern.  From  Sep- 
tember 21  to  December  21,  the  remainder  of  the  north 
frigid  zone  is  gradually  removed  from  the  sun's  light.  Dur- 
ing this  period,  day  continues  to  decrease  over  the  north- 
ern hemisphere,  and  to  increase  over  the  southern.  From 
December  21  to  March  21,  one-half  of  the  north  frigid 
zone  is  gradually  brought  again  into  the  sun's  light.  Dur- 
ing this  period,  day  is  increasing  over  the  northern  hemi- 
sphere, and  decreasing  over  the  southern.  From  March  21 
to  June  21,  the  remaining  half  of  the  north  frigid  zone  is 
brought,  by  the  earth's  motion,  into  the  sun's  light.  Day, 
therefore,  continues  to  increase  over  the  northern  hemi- 
sphere, and  to  decrease  over  the  southern. 

We  are  thus  enabled  to  see  that  the  northern  and  south- 
ern hemispheres  of  the  earth's  surface  alternate  with  each 
other  as  regards  their  situation  with  respect  to  the  sun's 
rays ;  which  alternation  is  a  necessary  result  of  the  con- 
stant parallelism  of  the  earth's  axis  during  its  revolution 
around  the  sun.  Turn  again  to  Fig.  1 1  for  an  illustration  of 
this.  The  upper  end  of  the  pencil  points  away  from  the 
centre  of  the  circle  in  one  position,  and  towards  it  in  the 
opposite  position :  the  reverse,  of  course,  is  true  with  re- 
gard to  the  lower  end.  Between  these  two  positions,  the 
pencil  is  perpendicular  to  the  radius  of  the  circle.  As  re- 
gards this  centre,  therefore,  the  pencil  moves  its  upper  and 
its  lower  half  reciprocally  to  and  from  it,  there  being  a 
position  midway  in  this  movement,  in  which  the  halves  are 
equally  near  to  the  centre  of  the  circle. 

It  is  also  evident,  from  Fig.  12,  that,  at  any  one  time,  the 
nearer  the  parallel  is  to  the  pole,  or  the  greater  the  latitude, 

3 


34  NEW    METHOD    OF 

the  greater  will  be  the  difference  in  the  lengths  of  day  and 
night. 

71.  We  now  learn  why  the  tropics  and  polar  circles  ajre 
distinguished  from  the  other  parallels  upon  the  earth's  sur- 
face :  for  the  former  are  the  farthest  parallels  from  the 
equator,  which   are  illuminated  by  a  vertical   sun  during 
the  year ;  and  the  latter  are  the  farthest  parallels  from  the 
poles,  which  pass  wholly  out  of,  or  wholly  into,  the  sun's  light. 

72.  When  a  parallel  lies  partly  in  the  illuminated  hemi- 
sphere and  partly  in  the  unilluminated,  or  is  divided  by 
the  day-circle,  it  has  a  day  and  a  night  every  24  hours, 
or   during   every  revolution  of   the    earth   upon    its    axis. 
When  it  lies  wholly  in  one  of  these  hemispheres,  its  day  or 
its  night  continues  until  it  is  again  divided  by  the  day-circle. 
Now,  as  we  have  seen,  parallels  of   the  frigid  zones  are 
the  only  parallels,  which,  at  certain  times  in  the  course 
of  a  year,  are  divided,  and  at  other  times  are  not  divided, 
by  the  day-circle.    These  parallels  are,  therefore,  subject  to 
a  greater  variety  of  day  and  night,  as  regards  length,  than 
occurs  within  the  temperate  and  torrid  zones.     They  have, 
indeed,  four  distinctly  marked  periods   during  the  year; 
these  periods  varying  in  length  according  to  the  distance 
of  the  given  parallel  from  the  pole. 

Naming  these  periods  in  the  most  convenient  way,  they 
are,  ist,  a  period  of  continuous  day,  during  which  the  paral- 
lel is  wholly  in  the  illuminated  hemisphere  ;  2dly,  a  period 
of  alternate  day  and  night,  during  which  the  parallel  is 
partly  in  the  illuminated  and  partly  in  the  unilluminated 
hemisphere  ;  3dly,  a  period  of  continuous  night,  during 
which  the  parallel  is  wholly  in  the  unilluminated  hemi- 
sphere ;  and  4thly,  a  second  period  of  alternate  day  and 
night,  during  which  the  parallel  is  again  partly  in  the  illu- 
minated and  partly  in  the  unilluminated  hemisphere. 

The  middle  of  a  period  of  continuous  day,  for  either  the 
northern  or  the  southern  hemisphere,  is  at  the  summer 
solstice  for  the  corresponding  hemisphere.  The  middle  of 


MOUNTING   AND    OPERATING   GLOBES.  35 

the  succeeding  period  of  alternate  day  and  night  is  at  the 
autumnal  equinox.  The  middle  of  a  period  of  continuous 
night  is  at  the  winter  solstice.  The  middle  of  the  suc- 
ceeding period  of  alternate  day  and  night  is  at  the  vernal 
equinox. 

73.  As  an  instance  of  this  variety  of  day  and  night  within 
the  frigid  zones,  let  us  see  how  it  is  exhibited  at  Spitz- 
bergen    during    the    year.      Day   gradually   increases    in 
length,  from  a  momentary  glimpse  of  the  sun  on  Febru- 
ary 21  to  12  hours  on   March  21  ;  then  to  24  on  April  21, 
when  it  remains  continuous  until  August  21 :  it  then  alter- 
nates with  night,  decreasing  from  24  hours  to  12  on  Sep- 
tember 21,  and  to  a  parting  glimpse  of  the  sun  on  October 
21  ;  when  a  continuous  night  of  four  months  succeeds. 

Farther  south,  as  in  the  southern  part  of  Nova  Zembla, 
we  should  find  a  continuous  day  and  night  of  about  6 
weeks  each,  and  periods  of  alternate  day  and  night  of  20 
weeks  each.  Farther  north,  on  the  contrary,  we  should  find 
that  the  periods  of  alternate  day  and  night  are  shorter, 
until,  at  the  pole,  they  cease  altogether,  and  the  two  pe- 
riods of  continuous  day  and  of  continuous  night,  each  six 
months  in  length,  compose  the  year. 

The  greatest  length  of  day  within  the  torrid  zone  is 
about  13^  hours,  this  length  occurring  upon  either  tropic. 
The  greatest  length  of  day  within  the  temperate  zones 
obtains  upon  the  polar  circles,  where  it  is  24  hours.  The 
length  of  any  given  day  subtracted  from  24  hours  gives 
the  length  of  the  night,  and  vice  versa. 

74.  Owing  to  the  variable  rate  at  which  the  sun  moves 
between  the  tropics,  the  relative  length  of  day  and  night, 
at  the  same  place,  also  changes  at  a  variable  rate.    Thus  this 
change  proceeds  the  slowest  at  the  times  of  the  solstices, 
and  the  fastest  at  the  times  of  the  equinoxes.     The  reason 
of  this  is  to  be  found  in  the  varying  inclination  of  the  sun's 
motion  along  the  ecliptic  to  the  equator.     Thus,  for  some 
time  before  and  after  the  solstices,  the  sun  is  describing 


36  NEW    METHOD    OF 

an  arc  of  the  ecliptic  (40°  or  50°  in  length),  which  is 
nearly  parallel  to  the  equator  on  both  sides  of  the  sol- 
stitial points  ;  consequently  the  change  in  the  sun's  declina- 
tion during  this  period  is  very  small.  During  this  period, 
therefore,  the  sun  describes  diurnal  circles  which  nearly 
coincide,  and  the  length  of  day  is  nearly  constant.  On  the 
other  hand,  about  the  time  of  the  equinoxes,  the  sun's  mo- 
tion has  the  greatest  inclination  to  the  equator ;  and,  there- 
fore, the  change  in  the  length  of  day  is  most  rapid.  The 
following  is  an  estimate  of  the  rate  at  which  its  declination 
increases  from  the  time  of  the  vernal  equinox.  From 
March  21  to  April  21,  the  sun  moves  northward  about 
10°;  from  April  21  to  May  21,  about  9°;  and  from  May 
21  to  June  21,  about  4°.  The  same  cause  also  affects,  to 
the  same  extent,  the  rate  at  which  the  sun  moves  to  and 
from  the  zenith  between  successive  noondays,  and  also  de- 
termines the  rate  at  which  it  advances  along  the  horizon 
between  successive  sunrises  or  sunsets. 

75.  It  follows,  from  the  equality  in  the  rate  at  which  the 
sun  is  changing  in  declination  at  dates  equally  removed 
from  either  solstice,  that  the  earth's  surface  is  illuminated 
precisely  in  the  same  manner  on  any  pair  of  such  days. 
Hence  for  every  day  in  the  year,  except  those  two  which 
date  at  the  solstices,  or  the  longest  and  the  shortest  day, 
there  corresponds  another  day  of  the  year  equal  to  it  in 
length;   and   hence,   also,   at   dates  equally   distant  from 
either  equinox,  the  sum  of  the  lengths  of  the  two  days  (or 
of  the  two  nights)  must  just  equal  24  hours. 

76.  Every  place  upon  the  earth's  surface   has  6  months 
of  day  during  the  year,  and  6  months  of  night.     At  the 
poles,  the  year  is  divided  into  a  day  and  a  night  of   6 
months  each;  at  the  equator,  every  24  hours  is  divided 
equally  into  a  day  and  a  night.     At  places  between  these 
two  positions,  the  sum  total  of  the  length  of  day  and  night 
is  known  from  this :  For  every  day  shorter  than   1 2  hours 
during  the  year,  there  corresponds  one  just  as  much  longer; 


MOUNTING   AND    OPERATING   GLOBES.  3/ 

and  the  same  is  true  of  night  •  making  the  average  length  of 
each  12  hours  for  the  year  ;  in  which  time  the  total  length  of 
each  must,  of  course,  sum  up  to  6  months,  as  at  the  equator. 

THE   DIURNAL   COURSE   OF   THE   SUN. 

77.  The  varied  changes  in  the  length  of  day  and  night 
may  be  also  studied  to  advantage  by  examining  the  diurnal 
course  of  the  sun  with  reference  to  the  horizon  of  a  place 
during  the  year. 

The  daily  motion  of  the  earth  upon  its  axis  from  west 
to  east  causes  an  apparent  motion  of  the  sun  across  the  sky 
from  east  to  west.  Now,  it  is  evident  that  the  apparent  mo- 
tion must  be  at  the  same  rate  as  the  real ;  that  is,  the  sun 
moves  both  above  and  below  the  horizon  at  the  rate  of  15° 
every  hour,  or  i°  every  4  minutes.  This  motion  is  per- 
formed in  a  great  circle  (viz.,  the  celestial  equator)  when 
the  sun  is  vertical  at  the  equator,  or  upon  March  21  and 
September  21 :  at  all  other  times,  the  diurnal  circles  de- 
scribed by  the  sun  are  small  circles.  An  observer  at  the 
north  pole  upon  March  21  or  September  21  would  be 
able  to  follow  the  sun's  course  completely  round  the 
celestial  equator,  which,  at  the  pole,  coincides  with  the 
horizon  (see  Fig.  7).  In  fact,  at  the  pole,  the  sun  is 
rising  upon  the  former  date,  and  setting  upon  the  lat- 
ter. If  he  continued  to  observe  this  course  for  three 
months  subsequently  to  March  21,  he  would  find  that  the 
diurnal  circles  described  by  the  sun  gradually  decreased  in 
size,  just  as  parallels  decrease  from  the  equator  to  the 
tropics.  The  sun's  diurnal  course  is  not,  however,  an 
exact  circle ;  since,  while  describing  this  course,  it  is 
changing  in  declination  :  it  follows,  in  fact,  the  direction 
of  a  spiral. 

The  sun  ascends  about  10°  above  a  polar  horizon  during 
the  first  month  of  the  long  polar  day,  9°  during  the  second, 
and  4°  during  the  third;  attaining  a  final  distance  of  23^° 
above  it,  after  which  it  descends  at  a  corresponding  rate. 


38  NEW    METHOD    OF 

This  motion  of  the  sun  from  and  to  the  horizon  is  identical 
with  its  change  in  northern  declination. 

The  points  of  the  compass  are  virtually  abolished  at  the 
poles ;  the  whole  horizon  being  in  the  direction  of  south 
at  the  north  pole,  and  of  north  at  the  south  pole. 

78.  At  the  equator  the  sun  always  describes  one-half  of  its 
diurnal  circle  above  the  horizon,  and  the  other  half  below  it. 
(See  Fig.  6,  illustrating  a  right  sphere,  page  18.)     Upon 
March  21  and  September  21  the  sun  rises,  in  the  east  and 
sets  in  the  west,  not  only  at  the  equator,  but  also  at  all 
places  on  the  earth's  surface,  unless  we  except  the  poles ; 
for  the  diurnal  course  of  the  sun  on  these  dates  coincides 
with  the  celestial  equator,  which,  as  shown  in  Art.  44,  inter- 
sects the  horizon  of  every  place  in  the  east  and  west  points. 

Subsequently  to  March  21  the  sun  rises  and  sets  north 
of  the  east  and  west  points  upon  an  equatorial  horizon, 
rising  each  successive  day  a  little  farther  from  the  east, 
and  setting  a  little  farther  from  the  west,  until  on  June  2 1 
it  rises  23!°  N.  of  E.,  and  sets  23^°  N.  of  W.  After  this 
it  begins  to  return  towards  the  east  and  the  west  points 
when  rising  and  setting. 

Subsequently  to  September  21  the  sun  rises  S.  of  E. 
and  sets  S.  of  W.  upon  an  equatorial  horizon,  until,  on 
December  21,  it  has  attained  a  distance  of  23^°  in  these 
directions ;  after  which,  in  the  course  of  the  succeeding 
three  months,  it  returns  to  the  east  and  west  points  once 
more,  reaching  them  on  September  21. 

Upon  March  21  and  September  21  the  sun  is  vertical, 
or  in  the  zenith,  at  noon,  for  every  situation  upon  the  equa- 
tor. From  March  21  to  September  21  it  culminates,  or 
comes  to  its  noon  position  north  of  the  zenith ;  and  during 
the  remainder  of  the  year  south  of  the  zenith,  attaining  its 
greatest  distance  from  the  zenith,  or  23^°  at  the  times  of 
the  solstices.  It  then  returns  towards  the  zenith,  reach- 
ing it  at  the  time  of  either  equinox. 

79.  The  horizon  of  a  place  between  the  equator  and  the 


MOUNTING   AND    OPERATING   GLOBES.  39 

poles  (see  Fig.  8,  page  18)  divides  the  equator  only,  of  all 
the  circles  of  declination,  into  two  equal  parts :  therefore 
one-half  of  the  sun's  diurnal  course  is  described  above  it 
only  upon  March  21  and  September  21.  If  the  place  is 
north  of  the  equator,  northern  circles  of  declination  (and 
here  we  need  only  consider  those  to  which  the  sun's  course 
is  confined,  or  which  extend  as  far  as  the  Tropic  of  Can- 
cer) have  their  longer  arc  above  the  horizon,  and  shorter 
below  it ;  and  southern  circles  have  their  shorter  arc  above 
the  horizon,  and  longer  below  it.  When,  therefore,  the  sun 
is  north  of  the  equator,  the  longer  arcs  are  described  by 
the  sun  during  the  day,  and  the  day  is  more  than  twelve 
hours  long ;  and  when  south  of  the  equator,  the  shorter 
arcs  are  described,  and  day  is  less  than  twelve  hours  long. 
The  reverse  is  the  case  if  the  place  is  south  of  the  equator. 
The  greater  the  distance  from  the  equator,  the  greater  the 
difference  between  the  length  of  the  two  arcs  into  which  a 
circle  of  declination  is  divided  by  the  horizon  ;  hence  the 
greater  the  contrast  between  the  length  of  the  day  and 
the  night. 

The  more  oblique  the  sphere,  or  the  greater  the  latitude 
of  the  place,  the  more  does  its  horizon  differ  in  direction 
from  that  of  the  sun's  motion  in  declination :  hence  the 
longer  the  distance  measured  upon  the  horizon  which  cor- 
responds to  a  change  of  a  given  number  of  degrees  in  de- 
clination. In  high  latitudes  the  sun  must  therefore  rise 
and  set  at  a  more  rapid  rate  of  advance  along  the  horizon, 
and  also  approach'more  nearly  to  its  northern  and  south- 
ern points.  This  rate  of  advance  is  nevertheless  affected 
by  the  sun's  rate  of  movement  between  the  tropics,  being 
considerably  slower  about  the  time  of  either  solstice  than 
at  the  time  of  an  equinox. 

The  horizon,  whose  inclination  to  the  equator  is  23^°, 
has  the  sun  within  its  northern  or  its  southern  point  at 
the  respective  times  of  the  solstices ;  so  that  sunrise  and 
sunset  must  take  place  within  every  point  of  this  horizon 


4<D  NEW    METHOD    OF 

during  the  year,  and  one  diurnal  circle  be  described  above 
it  at  the  summer  solstice.  This  is  the  case  at  places  upon 
the  polar  circles. 

80.  At  places  within  the  polar  circles  the  sun  rises  and 
sets  at  every  point  upon  their  horizons  during  the  year ; 
passing  along  the  horizon  at  a  more  rapid  rate  the  nearer 
the  place  is  to  the  pole,  and  reaching  the  north  or  the 
south  point  previously  to  the  time  of  a  solstice.     The  time 
previously  depends  upon  the  distance  of  the  given  place 
from  the  polar  circle,  being  greater  with  an  increase  of  lati- 
tude.    As   the   latitude   increases,  the  greater,  therefore, 
must  be  the  number  of  entire  diurnal  circles  which  the 
sun  describes  continuously  above  the  horizon  during  the 
year  ;  and  this  agrees  also  with  the  increase  in  the  con- 
tinuance of  day  in  this  direction  (see  Art.  73).     The  angle 
between  these  circles  and  the  horizon  diminishes  towards 
the  poles,  until,  at  the  poles,  a  difference  of  direction  be- 
tween  the   two   ceases   altogether,    and   the    sun    moves 
either  in  the  horizon,  or  in  a  direction   parallel   to   the 
horizon. 

The  rate  at  which  the  sun's  diurnal  circles  ascend  above 
the  horizon,  or  descend  towards  it,  within  the  polar  regions 
(not  now  including  the  poles),  depends  upon  the  position 
of  the  sun  within  the  ecliptic  when  describing  them.  If 
the  day  is  two  months  or  less,  the  sun  is,  during  that  time, 
describing  an  arc  of  the  ecliptic  nearly  parallel  to  the  equa- 
tor, and  moves  northward  or  southward,  therefore,  at  its 
slowest  rate.  When  the  day  is  a  longer  -one,  the  sun  leaves 
and  returns  to  the  horizon  at  its  most  rapid  rate,  or  nearly 
so,  during  the  former  and  the  latter  portion  of  it. 

81.  As  an  instance  of  the  variety  which  the  sun's  diurnal 
circuits  exhibit  in  the  frigid  zones,  let  us  follow  the  sun's 
course  at  Spitzbergen  for  a  year,  beginning  with  the  dawn 
succeeding  a  period  of  continuous  night. 

The  sun  appears  in  the  southern  point  of  the  horizon  upon 
February  21,  and  immediately  sets  without  an  intervening 


MOUNTING   AND    OPERATING   GLOBES.  4! 

course.  The  next  day  it  describes  a  small  arc,  the  next  a 
longer  one,  and  so  on,  until  it  rises  in  the  east  and  sets  in 
the  west,  having  moved,  both  in  rising  and  in  setting, 
through  a  quarter  of  the  horizon  in  coming  to  these  equi- 
noctial points.  It  now  culminates  10°  above  the  horizon. 
Subsequently  to  March  2 1  it  describes  arcs  increasing  in 
length  until  April  21,  when  its  whole  circuit  is  brought 
above  the  horizon,  and  so  remains,  rising  higher  and 
higher,  until  the  time  of  the  summer  solstice.  At  this 
time  the  sun  comes  to  the  meridian  of  Spitzbergen  at 
a  distance  of  35°  above  the  south  point  of  the  horizon, 
and  13°  above  the  north;  the  difference  between  these 
numbers  (or  22°)  showing  the  obliquity  of  its  diurnal 
circles  to  the  horizon  of  Spitzbergen.  After  this  date  the 
sun  begins  slowly  to  descend,  until  it  sinks  below  the 
northern  point  of  the  horizon  for  a  moment  on  August 
2 1  ;  after  which  it  describes  arcs  gradually  decreasing  in 
length  until  it  has  left  the  visible  heavens  altogether  upon 
October  21. 

82.  The  zenith  distance  of  the  sun  at  noon  for  any  given 
place  north  of  the  equator  is  equal  to  the  difference  be- 
tween the  latitude  of  the  place  and  the  sun's  declination,  if 
the  latter  is  north  declination  ;  or  to  their  sum,  if  it  is  south 
declination.     The  meridian  altitude  of  the  sun  is  always 
equal  to  the  difference  between  its  zenith  distance  and  90°. 
The  reason  for  these  rules  should  be  found  by  the  learner ; 
and  he  should  also  derive  a  corresponding  rule  for  places 
in  the  southern  hemisphere.     Thus,  when  vertical  at  the 
equator,  the  sun  culminates  20°  from  the  zenith,  or  70° 
above  the  horizon,  at  the  latitude  of  20°  N.  or  S.,  towards 
the  southern  point  of  the  horizon  at  the  former  latitude, 
and  northern  at  the  latter.     When  vertical  at  the  Tropic 
of  Capricorn,  its  zenith  distance  is  23^°  at  the  equator, 
33  i°  upon  the  loth  northern  parallel,  67^°  at  the  south 
pole,  and  so  on. 

83.  Whenever  the  sun  is  vertical  at  a  latitude  north  of  a 


42  NEW   METHOD    OF 

given  place,  it  culminates  at  a  point  of  the  meridian  north 
of  the  horizon  of  that  place ;  whenever  it  is  vertical  at  a 
latitude  south  of  a  given  place,  it  culminates  south  of 
the  horizon  of  the  place.  It  therefore  culminates  both 
upon  the  north  and  the  south  sides  of  the  zenith  at 
places  within  the  torrid  zones,  but  only  upon  one  side  of 
it  at  places  within  the  temperate  zones ;  namely,  always 
towards  the  southern  point  of  the  horizon  north  of  the 
Tropic  of  Cancer,  and  always  towards  the  northern  point 
of  the  horizon  south  of  the  Tropic  of  Capricorn. 

Beyond  the  torrid  zone  the  sun  culminates  at  its  ex- 
treme zenith  distances  at  the  times  of  the  solstices ;  occu- 
pying its  nearest  position  to  the  zenith  at  noonday  of  the 
summer  solstice,  and  its  farthest  position  from  the  zenith 
at  noonday  of  the  winter  solstice.  Within  the  torrid  zone 
it  culminates  at  its  farthest  position  from  the  zenith  at  the 
time  of  the  winter  solstice;  this  being  December  21  at 
places  north  of  the  equator,  and  June  2 1  at  places  south  of 
the  equator. 

MEASUREMENT   OF   TIME. 

84.  Any  series  of  events  which  take  place  at  equal  inter- 
vals of  time  may  be  employed  as  a  measure  of  time.     Thus 
the  flowing  of  sand  in  the  hour-glass,  the  vibrating  of  a 
pendulum,  and  the  revolution  of  a  star,  have  been  sever- 
ally used  for  this  purpose.     But   the  great   standard  of 
time  is  the  period  of  the  revolution  of  the  earth  on  its  axis, 
which,  by  the   most   exact  observations,  is  found  to  be 
always  the  same. 

85.  The  passage  of  a  heavenly  body  across  the  meridian 
of  an  observer  is  called  the  transit  of  the  body :  the  tran- 
sit nearer  the  zenith  of  the  observer  is  the  upper  transit ; 
that  more  remote  from  his  zenith  is  the  lower  transit. 

86.  A  sidereal  day  is  the  interval  of  time  between  two 
successive  upper  transits  of  a  star,  or  of  any  fixed  point  of 
the  celestial  sphere  over  the  same  meridian.     The  transits 


MOUNTING   AND    OPERATING   GLOBES. 


43 


Fig.  13- 


of  the  vernal  equinox  (see  Art.  54)  are  always  employed  to 
mark  the  beginning  and  end  of  the  sidereal  day. 

87.  A  solar  day  is  the  interval  of  time  between  two  suc- 
cessive upper  transits  of  the  sun  over  the  same  meridian. 
Had  the  earth  only  a  rotation  on  its  axis,  so  that  the  sun 
was  stationary  in  the  heavens  like  a  fixed  star,  the  solar 
and  the  sidereal  days  would  be  equal.    While,  however,  the 
diurnal  motion  of  the  earth  is  performed  once,  its  motion 
in  its  orbit  carries  it  eastward  nearly  i°  :  consequently  it 
must  perform  i°  more  than  a  rotation  before  the  same 
meridian  returns  to  the  sun,  or  between   two   successive 
upper  transits  of  the  sun.      Thus,  in  the  figure,  the  me- 
ridian A  B,  which  is  in  conjunc- 
tion with  the  sun  in  the  position 
of  the   earth   marked  i,  differs 
from  conjunction   with  the   sun 
by  the  arc  C  A  after  the  earth 
has  performed  exactly  one  rota- 
tion, and   reached    the   position 
marked  2.     Therefore  the  solar 
day   must    exceed    the    sidereal 
by  the  time  required   to   rotate 
through  the  arc  C  A,  which,  as 
we  have   said,  is   about    i°,   so 
that   the    solar  day   is    about   4 
minutes  longer  than  the  sidereal 
day.     In  other  words,  if  the  sun 
and  a  fixed  star  cross  the  me- 
ridian   together,    at    their    next 
transits  the  sun  will  cross  the  meridian  about  4  minutes 
behind  the  star.     When  this  has  gone  on  for  a  whole  year, 
the  sun  will  have  fallen  behind  the  star  by  a  whole  circum- 
ference of  the  heavens ;  that  is,  the  sun  will  have  made 
fewer  diurnal  revolutions  by  one  than  the  star.     Thus  the 
number  of  solar  days  in  the  year  must  be  one  less  than  the 
number  of  sidereal  days. 


44  NEW   METHOD    OF 

88.  The  solar  days,  however,  do  not  always  differ  from 
the  sidereal  by  precisely  the  same  amount ;  since,  owing  to 
the  unequal  motion  of  the  sun  in  the  ecliptic,  these  days 
are  not  constantly  of  the  same  length.     Time,  as  measured 
by  the  sun,  is  called  apparent  time.     Mean  time  is  time 
reckoned   by   the   average  length   of   all   the   solar   days 
throughout  the  year :  thus  the  mean  solar  day  is  the  av- 
erage length  of  the  solar  day  for  the  whole  year.     This 
period  constitutes  the  civil  day  of  twenty-four  hours,  be- 
ginning at  midnight,  or  when  the  sun  is  on  the  meridian 
at   its   lower  transit.     It  is  divided  into  two  periods  of 
twelve  hours  each,  from   the   lower  to   the  upper  transit 
(midnight  to  noon),  and  from  the  upper  to  the  lower  (noon 
to  midnight).     The  astronomical  day  is  the  solar  day  (ap- 
parent or  mean)  counted  through  the  whole  twenty-four 
hours  instead  of  by  periods  of  twelve  hours  each,  and  be- 
gins at  noon. 

89.  The  interval  by  which  apparent  time  differs   from 
mean  time  is  called  the  equation  of  time.    The  sun's  change 
of  right  ascension  is  sometimes  faster  than  if  it  moved  on 
the  equator,  and  sometimes  slower :  therefore  the  equation 
of  time  must  sometimes  be  added  to  apparent  time,  and 
sometimes  subtracted  from  it,  to  give  the  mean  time.     Its 
greatest   additive  value   is   about    14 \  minutes,  and   oc- 
curs about  February  u  ;  and  its  greatest  subtractive  value 
is  about  16^  minutes,  and  occurs  about  November  3.    The 
equation  of  time  is  zero,  or  mean  and  true  time  are  the 
same  four  times  in  the  year  ;  viz.,  about  April  15,  June  15, 
September  i,  and  December  24. 

Local  time  is  the  time  of  day  (solar  or  sidereal)  on  any 
given  meridian  at  any  instant.  Since  the  beginning  of  the 
day  is  determined  by  the  transit  across  the  meridian  of  the 
place  of  a  body  (sun  or  star)  which  has  an  apparent  diur- 
nal motion  at  the  rate  of  15°  per  hour,  it  follows  that  the 
local  time  at  two  places  situated  on  different  meridians 
must  differ  for  the  same  absolute  instant  of  time,  and  that 


MOUNTING   AND    OPERATING    GLOBES.  45 

a  difference  of  15°  in  longitude  must  correspond  to  a  dif- 
ference of  one  hour  in  time.  And,  since  the  apparent 
motion  of  the  sun  is  from  east  to  west  through  the  heavens, 
it  follows  that  the  local  time  is  later  at  the  more  easterly 
of  the  two  meridians.  Hence  a  watch  or  chronometer, 
which  is  set  to  the  time  of  any  meridian,  will  appear  to 
gain  if  carried  to  the  west,  and  to  lose  if  carried  to  the  east ; 
the  amount  gained  or  lost  in  any  case  being  equal  to  the 
difference  in  longitude  of  the  two  places  reduced  to  time. 
For  example :  if  a  watch  keeping  Boston  time  is  taken  to 
New  York,  it  will  be  found  to  be  about  1 2  minutes  fast  as 
compared  with  New- York  time.  This  indicates  that  the 
difference  in  longitude  of  the  two  places  is  about  3°,  or 
that  New  York  is  about  3°  west  of  Boston  in  longitude. 
This  example  illustrates  the  principle  by  which  the  longi- 
tudes of  places  are  actually  ascertained. 

THE   SEASONS. 

90.  The  perpetually  recurring  changes  of  the  seasons, 
modifying  the  organization  and  distribution,  as  well  as 
affecting  in  numberless  ways  the  welfare,  of  all  living 
things  on  the  earth,  may  be  easily  shown  to  result  from 
the  obliquity  of  the  ecliptic,  or,  what  comes  to  the  same 
thing,  the  inclined  position  of  the  earth's  axis  with  refer- 
ence to  the  plane  of  the  ecliptic.  From  this,  as  a  primary 
cause,  it  follows,  as  we  have  seen,  first,  that  the  length  of 
time  that  the  sun  remains  above  the  horizon  of  any  place 
varies  during  the  year  ;  secondly,  that  the  meridian  or  noon 
altitude  attained  by  the  sun  undergoes  corresponding  varia- 
tions. Now,  these  two  circumstances,  operating  as  sec- 
ondary causes,  regulate  the  supply  of  heat,  which,  along 
with  the  light,  the  earth's  surface  receives  from  the  sun  ; 
and  the  variations  in  this  supply  of  heat  determine  the 
changes  of  the  seasons. 

91.  The  effect  of  the  duration  of  the  day  upon  the  supply 
of  heat  is  quite  obvious.  Whenever  the  sun  is  above  the 


46  NEW    METHOD    OF 

horizon  of  a  place,  that  place  is  receiving  heat  from  it  ; 
when  below,  it  is  losing  heat  by  a  process  called  radiation, 
illustrated  every  time  a  teakettle  is  removed  from  the  fire, 
or  a  person  goes  out  of  a  warm  room  into  the  cold  air. 
The  longer  the  day,  therefore,  and  the  shorter  the  night, 
the  greater  the  supply  of  heat  from  the  sun  by  day, 
and  the  less  the  amount  lost  by  radiation  during  the 
night.  So  far,  then,  as  the  mere  length  of  the  day  is  con- 
cerned, it  is  clear  that  the  influence  of  the  sun  in  raising 
the  temperature  of  the  surface  of  the  earth  at  any  place  is 
greatest  when  the  days  are  longest  (i.  e.,  in  June,  for  the 
northern  hemisphere),  and  is  least  when  the  days  are  short- 
est, or  at  the  time  of  the  winter  solstice. 

This  cause  must  more  sensibly  affect  places  far  removed 
from  the  equator,  because  at  such  places  the  days  are 
longer  and  the  nights  shorter  than  in  the  torrid  zone.  By 
the  operation  of  this  cause,  the  solar  heat  accumulates 
there  so  much  during  the  longest  days  of  summer,  that 
the  temperature  rises  to  a  higher  degree  than  is  often  ex- 
perienced in  much  lower  latitudes. 

92.  But,  in  general,  the  second  cause  —  the  greater  or 
less  obliquity  of  the  sun's  rays  —  has  much  more  influence 
in  changing  the  temperature  of  the  earth's  surface  than  the 
mere  length  of  time  that  the  sun  is  above  the  horizon. 

"  The  higher  the  sun  ascends  above  the  horizon,  the  more 
directly  his  rays  fall  upon  the  earth ;  and  their  heating 
power  is  rapidly  augmented  as  they  approach  a  perpen- 
dicular direction.  When  the  sun  is  nearly  overhead,  his 
rays  strike  us  with  far  greater  force  than  when  they  meet 
us  obliquely ;  and  the  earth  absorbs  a  far  greater  number 
of  those  rays  of  heat  which  strike  it  perpendicularly  than 
of  those  which  meet  it  in  a  slanting  direction.  When  the 
sun  is  near  the  horizon,  his  rays  merely  glance  along  the 
ground ;  and  many  of  them,  before  they  reach  it,  are  ab- 
sorbed and  dispersed  in  passing  through  the  atmosphere. 
Those  who  have  felt  only  the  oblique  solar  rays,  as  they 


MOUNTING   AND    OPERATING   GLOBES. 


47 


Fig.  14. 


^//mV^S^X 


xw^>>; 


fall  upon  objects  in  the  high  latitudes,  have  a  very  inade- 
quate idea  of  the  power  of  a  vertical,  noonday  sun,  as  felt 
in  the  region  of  the  equator." 

The  exact  manner  in  which  the  inclination  of  the  surface 
of  the  earth  to  the  sun's  rays  modifies  the  supply  of  heat 
received  is  illustrated 
in  Fig.  14.  Let  A  B 
represent  a  portion 
of  the  earth's  surface 
upon  which  the  rays 
of  the  sun  (S)  fall  ver- 
tically, and  AC  an 
equal  surface  upon 
which  the  rays  fall  ob- 
liquely ;  and  suppose, 
further,  that  the  angle 
B  A  C  is  such  that  the 
ray  S  C  bisects  A  B. 
Now,  as  shown  in  the 
figure,  the  rays  impin- 
ging on  one-half  of  the  surface  A  B  would  be  spread  out 
over  the  entire  surface  A  C,  so  that  this  surface  would 
receive  only  one-half  as  much  heat  as  was  received  by  the 
equal  surface  A  B. 

93.  Both  of  these  causes  co-operate  in  the  temperate 
zones  to  elevate  the  temperature  of  the  earth's  surface  ;  for, 
when  the  days  are  the  longest,  the  meridian  altitude  of  the 
sun  is  the  greatest.  The  greatest  temperature  of  a  place, 
however,  is  not  at  the  time  of  the  summer  solstice,  but  about 
a  month  later,  because  the  amount  of  heat  received  by  the 
earth  during  the  day  continues,  for  some  time  after  the  sol- 
stice, to  exceed  the  amount  radiated  by  night :  consequently, 
heat  must  accumulate  until  these  amounts  become  equal. 
Afterwards,  the  decrease  being  greater  than  the  increase, 
the  heat  will  diminish.  For  a  similar  reason,  the  coldest 
period  occurs  some  time  after  the  winter  solstice. 


48  HEW    METHOD    OF 

94.  Since  the  earth  is  not  always  at  the  same  distance 
from  the  sun,  but  is  nearest  the  sun  at  the  perihelion  point, 
and  farthest  at  the  aphelion  point,  it  might  be  supposed 
that  this  circumstance  would  have  much  to  do  with  pro- 
ducing the  changes  of  the  seasons.     The  nearer  the  earth 
is  to  the  sun,  the  greater  must  be  the  intensity  of  the  solar 
heat,  and  vice  versa.     But  it  can  be  shown,  that  by  reason 
of  the  small  amount  of  the  variation  in  distance,  and  the 
unequal  rate  at  which  the  earth  moves  in  its  orbit,  this 
change  of  distance  has  but  a  very  trifling  influence  upon  the 
amount  and  distribution  of  light  and  heat  upon  the  earth. 

95.  The  seasons  vary  much  in  character  with  the  differ- 
ent zones  upon  the  earth's  surface. 

In  the  north  temperate  zone,  the  year  is  divided  into  the 
four  seasons,  —  spring,  summer,  autumn,  and  winter.  As- 
tronomically speaking,  these  begin  respectively  at  the  fol- 
lowing dates :  Vernal  equinox,  summer  solstice,  autumnal 
equinox,  winter  solstice.  Approximately,  these  dates  are 
March  21,  June  21,  September  21,  December  21  ;  but  the 
precise  dates  differ  somewhat  from  these,  owing  to  the  un- 
equal rate  of  the  motion  of  the  earth  round  the  sun,  and 
they  also  vary  within  small  limits  from  year  to  year. 

For  the  year  1872,  the  exact  dates  (in  mean  solar  time) 
of  the  equinoxes  and  solstices  are,  — 

Vernal  equinox,  March  20,  7  h.  6  m.  A.M. 
Summer  solstice,  June  21,  3  h.  41  m.  A.M. 
Autumnal  equinox,  September,  22,  6  h.  2  m.  P.M. 
Winter  solstice,  December  21,  0  h.  2  m.  P.M. 

Thus  winter  is  the  shortest  season,  and  summer  the 
longest,  while  spring  is  longer  than  autumn.  Spring  and 
summer,  taken  together,  are  about  eight  days  longer  than 
autumn  and  winter  combined. 

To  the  south  temperate  zone  what  has  been  said  of  the 
north  temperate  will  apply,  if  we  interchange  the  equi- 
noxes, the  solstices,  the  terms  winter  and  summer,  and 
the  terms  spring  and  autumn. 


MOUNTING   AND    OPERATING    GLOBES.  49 

In  the  torrid  zone,  the  motion  of  the  sun  with  respect  to 
the  zenith  is  different  from  what  it  is  in  the  temperate 
zones,  since  it  crosses  the  meridian  to  the  south  of  the 
zenith  during  part  of  the  year,  and  to  the  north  of  it  during 
the  remainder  of  the  year.  The  variations  in  the  length  of 
the  day  are  less  than  at  any  place  in  the  temperate  zones, 
and  the  average  noon  altitude  of  the  sun  during  the  year  is 
greater.  The  torrid  zone  is,  therefore,  a  region  of  great 
heat  throughout  the  year,  with  comparatively  little  differ- 
ence (so  far  as  temperature  is  concerned)  between  opposite 
seasons. 

The  frigid  zones  are  regions  of  low  temperature  on  the 
average  throughout  the  year,  and  of  excessive  winter  cold, 
contrasted  by  a  brief  seaspn  of  extreme  summer  heat. 

96.  Finally,  it  should  be  noticed  that  the  temperature  of 
a  place  is  influenced  very  much  by  local  and  temporary 
causes,  as  well  as  by  the  intensity  and  duration  of  the  sun's 
heat. 

"  First,  the  elevation  of  a  country  above  the  level  of  the 
sea  has  a  great  influence  upon  its  climate.  Elevated  dis- 
tricts of  country,  even  in  the  torrid  zone,  often  enjoy  the 
most  agreeable  climate  in  the  world.  The  cold  of  the  up- 
per regions  of  the  atmosphere  modifies  and  tempers  the 
solar  heat,  so  as  to  give  a  most  delightful  softness ;  while 
the  uniformity  of  temperature  excludes  those  sudden  and 
excessive  changes  which  are  often  experienced  in  less 
favored  climes.  In  ascending  certain  high  mountains  situ- 
ated within  the  torrid  zone,  the  traveller  passes  in  a  short 
time  through  every  variety  of  climate,  from  the  most  op- 
pressive and  sultry  heat  to  the  soft  and  balmy  air  of 
spring,  which  again  is  succeeded  by  the  cooler  breezes  of 
autumn,  and  then  by  the  severest  frosts  of  winter.  A 
corresponding  difference  is  seen  in  the  products  of  the 
vegetable  kingdom.  While  winter  reigns  on  the  summit 
of  the  mountain,  its  central  regions  may  be  encircled  with 
the  verdure  of  spring,  and  its  base  with  the  flowers  and 

4 


$O  NEW   METHOD   OF 

fruits  of  summer.  Secondly,  the  proximity  of  the  ocean 
also  has  a  great  effect  to  equalize  the  temperature  of  a 
place.  As  the  ocean  changes  its  temperature  during  the 
year  much  less  than  the  land,  it  becomes  a  source  of 
warmth  to  contiguous  countries  in  winter,  and  a  fountain 
of  cool  breezes  in  summer.  Thirdly,  the  relative  humidity 
or  dryness  of  the  atmosphere  of  a  place  is  of  great  im- 
portance in  regard  to  its  effects  on  the  animal  system.  A 
dry  air  of  ninety  degrees  is  not  so  insupportable  as  a  humid 
air  of  eighty  degrees ;  and  it  may  be  asserted  as  a  general 
principle,  that  a  hot  and  humid  atmosphere  is  unhealthy, 
although  a  hot  air,  when  dry,  may  be  very  salubrious.  In 
a  warm  atmosphere  which  is  dry,  the  evaporation  of 
moisture  from  the  surface  of  the  body  is  rapid,  and  its 
cooling  influence  affords  a  most  striking  relief  to  an  in- 
tense heat  without ;  but,  when  the  surrounding  atmosphere 
is  already  filled  with  moisture,  no  such  evaporation  takes 
place  from  the  surface  of  the  skin,  and  no  such  refreshing 
effects  are  experienced  from  this  cause.  Moisture  collects 
on  the  skin  ;  a  sultry,  oppressive  sensation  is  felt ;  and 
chills  and  fevers  are  usually  in  the  train."  (a) 

TWILIGHT. 

97.  Besides  being  necessary  for  respiration,  the  atmos- 
phere contributes  to  our  welfare  in  another  way  almost 
equally  important  The  molecules  of  which  it  is  composed 
reflect  in  every  direction  a  portion  of  the  light  which  they 
receive  from  the  sun,  and  these  numerous  reflections  pro- 
duce the  diffused  light  by  means  of  which  we  are  able  to 
see  objects  by  day  in  the  absence  of  the  direct  rays  of  the 
sun.  If  there  were  no  atmosphere,  every  place  which  was 
not  directly  illuminated  by  the  sun,  or  which  did  not  re- 
ceive rays  reflected  from  the  ground  or  other  objects, 
would  be  plunged  in  complete  darkness.  The  blue  color 

(a)  Olmstead's  Letters  on  Astronomy,  pp.  124-26. 


MOUNTING   AND    OPERATING   GLOBES.  51 

of  the  sky,  which  is  nothing  but  the  color  of  the  air  seen 
through  a  thickness  of  many  miles,  would  disappear,  and 
the  heavens  at  mid-day  would  be  uniformly  and  intensely 
black,  save  only  at  the  spots  occupied  by  the  sun,  moon, 
and  stars.  Moreover,  the  phenomenon  known  as  twilight 
would  have  no  existence.  The  instant  the  sun  disappeared 
from  view,  the  darkness  of  midnight  would  prevail,  and 
would  continue  until  abruptly  brought  to  an  end  by  the 
sun's  reappearance  above  the  horizon. 

The  season  of  twilight  which  we  enjoy  at  the  dawn 
and  the  close  of  day  is  due  to  the  power  of  the  atmos- 
phere to  reflect  and  scatter  the  solar  rays  which  it  receives. 
The  earth's  atmosphere  forms  a  shell  fifty  miles  or  there- 
abouts in  thickness  around  the  earth  :  hence  those  rays 
from  the  sun  which  are  nearly  tangent  to  the  earth's  sur- 
face must  pass  on  into  the  higher  regions  of  the  atmos- 
phere beyond,  and  from  these  they  are  reflected  into 
portions  of  the  atmosphere  which  would  otherwise  receive 
no  light  at  all.  This  reflected  light,  properly  speaking, 
constitutes  twilight  (doubtful  light,  as  the  word  means  in 
the  original  Anglo-Saxon).  At  any  given  place,  when  the 
sun  is  just  setting  or  rising,  the  entire  atmosphere  which 
lies  above  the  plane  of  the  horizon  is  directly  illuminated 
by  the  sun.  After  sunset,  the  portion  of  the  atmosphere 
directly  illuminated  constantly  diminishes,  and  with  it  also 
the  quantity  of  light  reflected  to  the  place  in  question, 
until  at  last  no  sunlight  enters  directly  the  region  of  the 
atmosphere  referred  to,  when  twilight  is  said  to  be  at  an 
end.  In  the  morning,  similar  changes  in  the  amount  of 
light  received  occur  in  an  inverse  order. 

In  Fig.  15,  let  A  represent  a  given  place,  at  which  the 
sun,  S,  is  in  the  act  of  setting ;  the  ray,  S  B,  representing 
a  plane  of  rays  tangent  to  the  earth  at  A,  and  coinciding 
with  the  plane  of  the  horizon  at  A,  and  therefore  including 
above  it  the  entire  visible  sky.  In  this  case,  A  receives 
light  from  the  whole  reflecting  atmosphere,  C  D  E  :  twilight 


52  NEW    METHOD    OF 

is,  in  fact,  just  beginning.  In  the  position  of  the  sun 
marked  S',  A  receives  light  from  only  the  portion  D  E  of 
the  atmosphere.  The  degree  of  illumination  at  A  depends 

Fig.  15. 


on  the  size  of  this  portion,  and  therefore  grows  less  as  the 
sun  recedes  from  the  horizon.  At  S"  the  sun's  rays  graze 
the  exterior  limit  of  the  section  C  D  E,  and  twilight  fades 
completely  away. 

Twilight,  astronomically  considered,  is  generally  said  to 
continue  while  the  sun  is  within  18°  of  the  horizon,  meas- 
uring in  the  direction  of  a  vertical  circle  (see  vertical  circle, 
Art.  43)  ;  but  this  distance  really  varies  from  15°  to  20°. 
Heat  tends  to  lessen  the  limit,  and  cold  to  increase  it. 

The  time  which  the  sun  occupies  in  moving  to  a  dis- 
tance of  1 8°  from  a  given  horizon  depends  both  upon  the 
angle  between  the  circle  of  declination  which  the  sun  is 
describing  and  the  horizon  of  the  place,  and  upon  the  size 
of  this  circle  of  declination.  The  effect  of  a  change  of  the 
angle  may  be  illustrated  by  supposing  two  hoops  to  cross 


MOUNTING   AND    OPERATING   GLOBES.  53 

each  other  at  various  angles.  Suppose,  now,  that  we  find  a 
point  on  one  of  the  hoops  (representing  the  sun's  course), 
which,  in  the  several  positions,  is  at  a  given  distance  from 
the  other  hoop  (which  represents  the  horizon) :  evidently, 
the  less  the  angle  between  the  hoops,  the  greater  the  dis- 
tance of  this  point  from  the  intersection  of  the  hoops. 

The  only  horizons  for  which  the  sun's  entire  diurnal 
course  is  at  any  time  coincident  with  a  vertical  circle  are 
those  of  places  upon  the  equator  at  the  times  of  the  equi- 
noxes. The  shortest  period  of  twilight,  therefore,  is  ex- 
perienced at  the  equator  on  March  21  and  September  21, 
and  is  about  i  hour  and  12  minutes.  At  all  other  times 
the  sun's  course  is  upon  a  circle  of  declination  parallel  to, 
but  smaller  than,  the  equator.  This  increases  slightly  the 
duration  of  twilight  for  places  on  the  equator ;  the  longest 
twilight  occurring  at  the  times  of  the  solstices,  and  being 
about  i  hour  and  19  minutes. 

The  greater  the  latitude  of  a  place,  the  greater  the 
obliquity  of  its  horizon  to  the  sun's  diurnal  course  :  there- 
fore there  is  a  constant  increase  in  the  duration  of  twilight 
as  we  proceed  from  the  equator  towards  the  pole.  There 
is  also  a  constant  change  in  the  duration  of  twilight  at  the 
same  place  during  the  year,  this  change  attending  the 
changes  of  points  on  the  horizon  in  which  the  sun  rises 
and  sets. 

The  varying  angle  at  which  the  sun  proceeds  with  re- 
spect to  a  given  horizon,  during  the  year,  is  very  simply 
shown  upon  the  globe  accompanying  this  manual  by  adjust- 
ing the  ring  arrangement  upon  this  globe  to  any  given  place, 
as  Boston,  and  then  comparing  the  angles  which  the  brass 
horizon  forms  upon  the  north  with  the  equator,  and  one  or 
two  small  circles,  as  the  tropics. 

The  direction  of  the  sun's  diurnal  motion  at  rising  and 
setting  for  places  north  of  the  equator  most  nearly  coin- 
cides with  that  of  a  vertical  circle  on  March  6,  or  when  the 
sun  describes  a  parallel  about  6°  south  of  the  equator: 


54  NEW    METHOD    OF 

therefore,  at  this  time,  places  in  north  latitude  have  their 
shortest  twilight.  On  the  other  hand,  the  direction  of  the 
sun's  motion  makes  the  smallest  angle  with  the  horizon  at 
the  time  of  the  summer  solstice :  therefore  places  in  north 
latitude  have  their  longest  twilight  upon  June  21.  Places 
in  south  latitude  have  their  shortest  twilight  upon  April  6, 
and  their  longest  upon  December  2 1 . 

When  a  parallel,  having  an  arc  in  day,  extends  into  the 
unilluminated  hemisphere  of  the  earth's  surface  for  a  dis- 
tance of  more  than  18°  beyond  the  day-circle,  it  has  morn- 
ing and  evening  twilight.  When  it  extends  only  18°,  or 
less,  it  has  a  twilight  continuing  through  the  night,  or  last- 
ing from  sunset  to  sunrise.  When  the  whole  parallel  is 
embraced  within  the  18°,  it  has  a  period  of  continual  twi- 
light. Stockholm  has  a  twilight  lasting  from  sunset  to 
sunrise  for  a  space  of  4  months.  The  85th  parallel  has  a 
continuous  twilight  of  about  28  days. 

Astronomical  twilight,  above  considered,  is  much  longer 
than  what  is  ordinarily  regarded  as  twilight. 


MOUNTING   AND    OPERATING   GLOBES.  55 


DESCRIPTION   OF   THE   GLOBE* 


THE  stand  of  this  globe  supports  upon  its  upper  sur- 
face a  metal  disk,  fitted  to  revolve  about  a  vertical  axis, 
and,  by  its  revolution,  to  represent  the  progress  of  the 
earth  in  its  orbit,  and  of  the  sun  in  the  ecliptic.  Around 
this  disk  are  marked,  in  two  concentric  circles,  the  names 
of  the  calendar  months,  (a)  and  the  signs  of  the  ecliptic ; 
each  month  and  sign  being  divided  into  30  equal  parts  by 
means  of  the  subdivisions  around  the  edge  of  the  cylinder, 
these  parts  representing  days  in  the  months,  and  degrees 
in  the  signs.  These  subdivisions  are  numbered  at  every 
loth  day  and  degree ;  the  outer  numbers  referring  to  days, 
and  the  inner  to  degrees.  An  index  pointing  towards  the 
disk,  upon  a  level  with  its  upper  surface,  records  the  pas- 
sage of  its  subdivisions  as  it  revolves.  This  index  is 
called  the  calendar  index,  and  the  disk  the  calendar  disk. 

Attached  to  the  calendar  disk,  and  making  with  it  an 
angle  of  66£°,  is  a  metal  rod,  or  axis,  representing  the  axis 
of  the  earth.  Upon  this  axis  a  common  terrestrial  globe  is 
mounted,  and  turns  freely.  The  axis  is  not  fastened  to 


(a)  These  months  are  uniformly  divided  into  30  days  each ;  it  being 
necesary  to  have  the  dates  of  the  equinoxes  and  solstices  exactly  90° 
apart  upon  the  calendar  disk,  in  order  that  the  solar  index  shall  be 
opposite  the  equator  or  a  tropic  when  these  dates  are  brought  to  the 
calendar  index. 


56  NEW    METHOD    OF 

the  centre  of  the  disk,  but  to  such  a  point  that  the  centre 
of  the  globe  shall  always  remain  vertically  over  the  centre 
of  the  disk,  while  the  latter  revolves  about  a  vertical  axis. 
In  the  course  of  one  revolution  of  the  disk,  the  axis  evi- 
dently describes  or  generates  the  surface  of  a  cone  whose 
vertex  coincides  with  the  centre  of  the  globe.  The  body 
of  the  globe  is  thus  made  to  occupy  the  same  position 
with  respect  to  the  stand  at  all  times ;  and  yet,  by  the  coni- 
cal motion  above  described,  its  axis,  together  with  its  sur- 
face, is  shifted  with  respect  to  any  fixed  line  passing 
through  its  centre,  through  a  maximum  distance  of  twice 
23l  degrees  during  each  semi-revolution  of  the  disk. 

Directly  above  the  calendar  index  is  another  index, 
pointing  towards  the  centre  of  the  globe,  called  the  solar 
index.  It  represents  the  sun,  or,  more  properly,  the  central 
ray  proceeding  from  the  sun ;  so  that  the  position  upon  the 
surface  of  the  globe  to  which  it  points  at  any  time  repre- 
sents the  place  upon  the  earth's  surface  at  which  the  sun 
is  vertical  at  that  time. 

The  divisions  of  day,  night,  and  twilight,  upon  the  earth's 
surface,  are  represented  upon  this  globe  by  means  t>f  two 
brass  circles  (these  circles  being  cut  away  beneath,  in  order 
to  afford  passage  to  the  axis  of  the  globe)  rising  from  a 
semicircular  support,  which,  with  the  circles,  admits  of 
removal  from  the  globe-stand.  When  adjusted  to  the 
globe-stand,  the  larger  circle  is  so  situated  that  every  point 
of  it  is  90°  distant  from  the  solar  index :  it  divides  the 
day-side  of  the  globe  from  its  night-side,  and  is  called  the 
day-circle.  The  smaller  circle  includes  between  it  and 
the  larger  one  the  section  embraced  by  twilight  upon  the 
earth's  surface,  and  divides  twilight  from  dark-night:  it 
is  called  the  twilight-circle ',  and  is  situated  at  a  distance 
of  1 8°  measured  upon  a  great  circle  from  the  day-circle. 

From  this  method  of  mounting,  it  follows  that  the  globe 
has  two  distinct  motions ;  namely,  a  revolution  along  with 
the  disk  about  a  vertical  axis,  and  a  rotation  alone  about 


MOUNTING   AND    OPERATING   GLOBES.  57 

an  axis  which  always  makes  with  the  plane  of  the  disk  the 
angle  of  66  £°.  The  former  motion  represents  the  yearly 
motion  of  the  earth,  and  the  latter  its  daily  motion ;  the 
movement  in  both  cases  being  made  from  left  to  right.  As 
the  disk  is  revolved,  the  rod  representing  the  earth's  axis 
assumes,  in  the  course  of  one  revolution,  every  position  with 
respect  to  the  solar  index,  or  rather  with  respect  to  the 
line  of  direction  of  this  index,  that  is  actually  assumed  by 
the  earth's  axis  relatively  to  the  line  joining  the  centres  of 
the  sun  and  the  earth  during  the  annual  revolution  of  the 
earth  around  the  sun.  The  solar  index  describes  the  pas- 
sage of  the  sun  between  the  tropics,  marking  out  the  entire 
circle  of  the  ecliptic  in  the  course  of  the  revolution.  The 
brass  circles  meantime  indicate  the  changes  of  length  in 
day,  night,  and  twilight ;  these  changes  being  more  fully 
illustrated  by  simply  rotating  the  globe  on  its  axis  for  dif- 
ferent times  in  the  year. 

The  calendar  index  always  points  to  the  day  of  the  year 
for  which  the  solar  index  and  brass  circles  are  in  position. 
Small  knobs  projecting  from  the  outer  surface  of  the  cal- 
endar *disk  serve  to  manipulate  it.  A  nut  on  the  under 
side  of  the  globe-stand  may  be  screwed  up,  should  the  disk 
move  too  easily. 

A  ring  arrangement  consisting  of  a  ring  called  the  brass 
horizon,  and  a  semi-ring  called  the  brass  meridian,  is 
employed  to  represent  the  horizon  and  meridian  of  any 
given  place.  It  is  fitted  to  position  on  the  globe  for  any 
place  by  adjusting  the  perforation  at  the  middle  of  the 
semi-ring  to  the  place,  with  the  brass  meridian  passing 
in  the  direction  of  the  meridian  of  the  place.  The  globe 
being  now  revolved  upon  its  axis,  the  solar  index  describes 
the  daily  course  of  the  sun  (if  the  sun  rises  and  sets  upon 
the  given  day)  by  moving  from  side  to  side  of  the  brass 
horizon,  crossing  the  brass  meridian  at  the  time  of  noon. 

When  the  ring  arrangement  is  adjusted  to  a  given  place, 
the  extremity  of  the  brass  meridian  north  of  the  equator 


58  NEW    METHOD    OF 

represents  the  northern  point  of  the  horizon,  and  that  south 
of  it  the  southern.  The  east  and  west  positions  on  the 
brass  horizon  are  each  indicated  by  a  black  zero  (o)  ;  the 
former  being  upon  the  right  of  the  brass  meridian,  and 
the  latter  upon  the  left.  Both  the  brass  meridian  and  hori- 
zon are  graduated  upon  their  outer  surfaces. 

When  the  ring  arrangement  of  this  globe  is  to  be 
brought  into  use,  the  brass  circles  must  first  be  removed ; 
which  is  done  by  drawing  upon  the  knob  which  pro- 
jects from  their  circular  base.  When  the  brass  circles  are 
required  for  use,  the  ring  arrangement  must  first  be  lifted 
from  the  globe  ;  this  being  easily  done  when  the  circles  are 
away.  The  larger  brass  circle  is  graduated  upon  its  day- 
side,  its  degrees  being  numbered  at  intervals  of  ten,  and 
corresponding  at  these  intervals  with  parallels  of  latitude 
upon  the  globe,  when  this  is  in  position  for  March  21. 

In  order  to  measure  in  degrees  arcs  of  any  great  circle 
on  the  globe,  as,  for  example,  vertical  circles,  or  the  dis- 
tance of  any  two  places  from  each  other,  a  cloth  quadrant 
graduated  to  degrees,  and  called  the  altitude  quadrant,  is 
provided.  This  quadrant  may  be  threaded  into  the  perfora- 
tion of  the  ring  arrangement  whenever  required  for  finding 
the  sun's  azimuth,  altitude,  &c. 


MOUNTING  AND  OPERATING  GLOBES.  59 

PART     II. 

PROBLEMS   UPON   THE  GLOBE. 


N.  B.  —  IN  practising  upon  the  globe,  set  its  stand  so 
that  the  side  upon  which  the  solar  index  is  situated  shall 
be  turned  towards  you.  Problems  XVI.,  XXXVI.,  and 
XXXVII.  call  into  use  the  ring  arrangement.  For  the 
remainder,  have  this  removed,  and  the  brass  circles  in 
place. 


PROBLEMS   UPON   LATITUDE,   LONGITUDE, 
AND   DISTANCE. 


PROBLEM  I. 

To  find  the  Latitude  and  Longitude  of  a  given  Place. 

1.  Bring  March  21  to  the  calendar  index. 

2.  Bring  the  given  place  to  the  graduated  edge  of  the 
day-circle,  upon  the  east. 

3.  Read  off  the  latitude  upon  the  day-circle,  over  the 
place  ;   and  the  longitude  upon  the  equator,  where  the 
graduated  edge  of  the  day-circle  cuts  it  upon  the  east. 

Exercises.  —  Find  the  latitude  and  longitude  of  Wash- 
ington. 

Ans.  —  Lat.  39°  N.,  long.  77°  W. 

Find  the  latitude  and  longitude  of  Amsterdam,  Berlin, 
Paris,  Boston,  St.  Louis,  Valparaiso. 


6O  NEW    METHOD    OF 

PROBLEM  II. 

Given  the  Latitude  and  Longitude,  to  find  the  Place. 

1.  Bring  March  21  to  the  calendar  index. 

2.  Bring  the  given  degree  of  longitude  to  the  graduated 
edge  of  the  day-circle,  upon  the  east. 

3.  Look  for  the  given  latitude  upon  the  side  of  the  day- 
circle  with  the  given  longitude  :  the  place  will  be  under 
this  latitude. 

Exercises.  —  Find  the  place  whose  longitude  is  68°  E., 
and  latitude  58°  N. 

Ans.  —  Tobolsk. 

Find  those  places  whose  latitudes  and  longitudes  are 
as  follows;  riamely,  52°  N.  lat.  and  5°  E.  long.,  31°  N. 
lat.  and  30°  E.  long.,  52°  N.  lat.  and  13°  E.  long.,  42°  N.  lat. 
and  70°  W.  long.,  6°  N.  lat.  and  52°  W.  long.,  12°  S.  lat. 
and  76°  W.  long. 

PROBLEM  III. 

To  find  all  those  Places  that  are  in  the  same  Latitude  or 
Longitude  with  a  given  Place. 

1.  Bring  March  21  to  the  calendar  index. 

2.  Bring  the  given  place  to  the  graduated  edge  of  the 
day-circle,  upon  the  east ;  and  all  those  places  under  the 
same  edge,  as  far  as  the  poles,  are  in  the  same  longitude. 

3.  Turn  the  globe  upon  its  axis ;  and  all  those  places 
passing  under  the  latitude  of  the  given  place,  as  marked 
upon  the  day-circle,  are  in  the  same  latitude. 

Exercises.  —  Find  all  those  places  that  are  in  the  same, 
or  nearly  the  same,  latitude  or  longitude  as  Cairo. 

Ans.  —  Alexandria,  Odessa,  and  St.  Petersburg  are 
nearly  in  the  same  longitude ;  Delhi,  Houston,  Mobile, 
and  Tallahassee  are  nearly  in  the  same  latitude. 


MOUNTING   AND    OPERATING    GLOBES.  6 1 

Find  all  those  places  that  are  in  the  same,  or  nearly  the 
same,  latitude  or  longitude  as  Paris,  Constantinople,  Lon- 
don, Lima,  San  Francisco,  Montreal,  New  York. 


PROBLEM  IV. 
Given  two  Places,  to  find  their  Difference  of  Latitude. 

FIRST    METHOD. 

1.  Find  the  latitude  of  both  places  by  Problem  I. 

2.  Count  the    degrees    between   the   two  latitudes   as 
marked  upon  the  day-circle. 

SECOND    METHOD. 

1.  Find  the  latitude  of  both  places  by  Problem  I. 

2.  Add  the  two  latitudes  if  upon  opposite  sides  of  the 
equator  ;   subtract  the  smaller  from  the  greater  if  upon 
the  same  side. 

Exercises.  —  Find  the   difference   of    latitude   between 
Guatemala  and  Stockholm.    • 


Find  the  difference  of  latitude  between  Philadelphia  and 
New  Orleans,  Paris  and  Madrid,  Quito  and  Calcutta,  Dub- 
lin and  Baltimore,  Havana  and  Batavia,  Jerusalem  and 
Stockholm. 


PROBLEM  V. 
Given  two  Places,  to  find  their  Difference  of  Longitude. 

FIRST    METHOD. 

1.  Find  the  longitude  of  both  places  by  Problem  I. 

2.  Count  the  degrees  between  them  as  marked  upon 
the  equator,  reckoning  upon  the  arc  which  measures  the 
shorter  distance  between  them. 


62  NEW    METHOD    OF 


SECOND    METHOD. 

1.  Find  the  longitude  of  both  places  by  Problem  I. 

2.  Add  the  two  longitudes  if  upon  opposite  sides  of  the 
meridian  of  Greenwich,  subtracting  their  sum  from  360  if 
it  exceed   180 ;  subtract  the  smaller  from  the  greater  if 
upon  the  same  side. 

Exercises. —  Find  the  difference  of  longitude  between 
Boston  and  Cairo. 

Ans. — 103°. 

Find  the  difference  of  longitude  between  Pekin  and 
Aden,  Bombay  and  Lisbon,  Madras  and  San  Francisco, 
Nankin  and  Mobile,  Rio  Janeiro  and  Gondar,  Damascus 
and  Cork. 


PROBLEM  VI. 

To  find  the  Antceci^  Period,  and  Antipodes  of  a  given  Place. 

Note.  —  The  antoeci  are  the  inhabitants  of  the  earth  who  live  in  the 
same  longitude,  but  in  opposite  latitudes,  —  the  one  part  as  far  north 
of  the  equator  as  the  other  south. 

The  perioeci  are  the  inhabitants  of  the  earth  who  live  in  the  same 
latitude,  but  in  opposite  longitudes. 

The  antipodes  are  the  inhabitants  of  the  earth  who  live  diametri- 
cally opposite  to  each  other. 

1.  Find  the  latitude  of  the  given  place  by  Problem  I., 
and  follow  along  the  day-circle  to  the  same  latitude  in  the 
opposite  hemisphere  for  the  antceci. 

2.  Find  the  longitude  of  the  given  place  by  Problem  I., 
and  follow  along  the  parallel  of  the  place  until  it  again 
intersects  the  day-circle  for  the  periceci. 

3.  Find  the  antceci  of  the  given  place,  and  follow  along 
the  parallel  of  the  antceci  until  this  parallel  again  inter- 
sects the  day-circle  for  the  antipodes. 

Or  find   the   periceci  of  the  given   place*  and  follow 


MOUNTING   AND    OPERATING   GLOBES.  63 

along  the  day-circle  to  the  same  latitude  in  the  opposite 
hemisphere. 

Note  i.  — The  learner  should  employ  both  methods  of  finding  the 
antipodes  of  a  place,  and  thus  satisfy  himself  that  they  both  give  the 
same  result. 

Note  2.  —  Those  places  situated  on  the  equator  have  no  antoeci, 
and  their  perioeci  are  their  antipodes.  The  poles  have  no  periceci, 
and  their  antoeci  are  their  antipodes. 

Exercises.  —  Find  the  antceci,  periceci,  and  antipodes 
of  Canton. 

Ans.  —  The  antceci  are  in  the  north-western  part  of  Aus- 
tralia, near  Sharks  Bay ;  the  periceci  are  midway  between 
the  Bermudas  and  the  Caribbees ;  and  the  antipodes  are 
in  the  northern  part  of  the  Argentine  Republic,  between 
the  Rivers  Grande  and  Salado. 

Find  the  antceci,  periceci,  and  antipodes  of  Mecca,  Cape 
Horn,  Valparaiso,  Lima,  Dresden,  Manilla. 


PROBLEM   VII. 

To  find  the  Distance  between  two  Places. 

1.  Lay  the  edge  of  the  altitude  quadrant  over  the  two 
places,  so  that  the  point  marked  zero  (o)  may  be  over  one  of 
them :  the  number  of  degrees  over  the  other  place  will  give 
the  number  of  degrees  that  they  are  apart. 

2.  Multiply  the  number  of  degrees  by  60  to  obtain  the 
distance   in  geographical   miles ;   by  69.  i  to  obtain  it  in 
statute  miles,  (a) 

Note.  —  If  the  two  places  are  upon  the  equator  or  the  same  merid- 
ian, the  distance  between  them  can  be  estimated  upon  the  circle  upon 
which  they  lie. 

(a)  A  degree  of  a  great  circle  of  the  earth  considered  as  a  sphere 
is  divided  into  60  geographical  miles,  and  contains  69.1  common  or 
statute  miles. 


64  NEW    METHOD    OF 

Exercises.  —  Find  the  distance  in  geographical  miles 
between  London  and  Botany  Bay.  (a) 

Ans.  — 9,240  miles. 

Find  the  distance  in  geographical  miles  between  Wash- 
ington and  London,  Sitka  and  Nankin,  Manilla  and 
Aden ;  in  statute  miles  between  Boston  and  New  Or- 
leans, Dublin  and  Cincinnati,  Cayenne  and  Valparaiso. 


PROBLEM   VIII. 

Given  the  Latitude  or  Longitude  of  a  Place,  and  its  Distance 
from  a  given  Place,  to  find  the  Place  whose  Latitude  or 
Longitude  is  given. 

1.  Reduce  the  given  distance  to  degrees  by  allowing  60 
geographical,  or  69.1  statute  miles  to  a  degree. 

2.  Place  zero  (o),  on  the  quadrant,  over  the  given  place. 

3.  Direct  the  quadrant  towards  the  east  or  the  west,  ac- 
cording to  the  direction  of  the  required  place  from  the 
given  place  ;  and  lay  the  degree  upon  it,  equalling  the  given 
distance,  over  the  given  latitude  or  longitude  :  beneath  it 
will  be  found  the  required  place. 

Example.  —  Find  the  place  in  latitude  32°  N.  which  is 
2,750  geographical  miles  south-west  from  the  Lizard,  in 
England. 

Ans.  The  Bermudas. 

Find  the  place  in  latitude  60°  N.  which  is  1,320  statute 
miles  north-east  from  London;  in  longitude  118°  W.  which 
is  2,766  statute  miles  south-west  from  Boston  ;  in  longitude 
10°  E.  which  is  3,596  statute  miles  north-east  from  Wash- 
ington ;  upon  the  equator  which  is  3,600  geographical 


(a)  When  the  distance  between  the  two  places  exceeds  the  length 
of  the  quadrant,  the  ring  arrangement  may  be  used  instead. 


MOUNTING   AND    OPERATING    GLOBES.  65 

miles  south-west  from  Cape  Verde ;  in  latitude  10°  S. 
which  is  4,710  miles  south-west  from  Hawaii ;  in  longitude 
135°  W.  which  is  3,420  geographical  miles  north-east  from 
Jeddo. 


PROBLEM   IX. 

To  find  the  Length  of  a  Degree  of  Longitude  corresponding  to 
a  given  Latitude. 

1.  Ascertain  how  many  degrees  upon  the  altitude  quad- 
rant are  equal  to  15°  of  longitude  at  the  given  latitude. 

2.  Reduce  this  number  of  degrees  to  miles  by  direction  2, 
Problem  VII.;  and  divide  the  product  by  15,  the  number 
of  degrees  measured  at  the  given  latitude.     Or,  — 

3.  Reduce  this  number  of  degrees  to  geographical  miles 
by  multiplying  by  4,  to  statute  miles  by  multiplying  by  4.6. 

Example.  —  Find  how  many  geographical  miles  make  a 
degree  of  longitude  at  the  latitude  of  Philadelphia. 

Ans.  —  48. 

Find  how  many  miles,  either  geographical  or  statute, 
make  a  degree  of  longitude  upon  the  parallels  of  15°  N., 
30°  N.,  60°  N.,  upon  the  parallels  of  Boston,  London, 
Paris. 

Note.  — The  number  of  miles  in  a  degree  of  longitude  upon  a  given 
parallel  is  proportional  to  the  rate  at  which  one  situated  upon  the 
parallel  is  carried  from  west  to  east  by  the  revolution  of  the  earth 
upon  its  axis. 

N.B.  —  In  working  the  three  preceding  problems,  the  ring  ar- 
rangement may  be  employed,  in  case  a  quadrant  is  not  at  hand ;  or 
a  thread  will  serve  the  purpose  very  well,  it  being  applied  to  the 
equator  to  reduce  any  required  portion  upon  it  to  degrees. 


66  NEW    METHOD    OF 


PROBLEMS   UPON   TIME. 


PROBLEM  X. 

To  bring  a  given  Place  into  a  given  Hour  of  the  Day  or 
Night. 

FIRST   METHOD. 

1.  Bring  the  given  place  into   12  o'clock,  or  noon,  by 
bringing  its  meridian  to  the  solar  index. 

2.  If  the  given  time  is  later  than  12,  turn  the  globe  upon 
its  axis  towards  the  east  until  as  many  intervals  of  longi- 
tude (of  15°  each)  have  passed  the  solar  index  as  the  time 
is  past  12,  reckoned  in  hours. 

3.  If  the  given  time  is  earlier  than  12,  turn  the  globe  upon 
its  axis  towards  the  west  until  as  many  intervals  of  longitude 
are   passed  by  the  solar  index  as  the  time  is  before  12, 
reckoned  in  hours. 

SECOND   METHOD. 

1.  Bring  the  meridian  of  the  given  place  to  the  solar 
index. 

2.  Set  the  hour-index  to  12. 

3.  If  the  given  time  is  later,  turn  the  globe  upon   its 
axis  towards  the  east  until  the  hour-index  points  to  this 
time. 

4.  If  the  given  time  is  earlier,  turn  the  globe  upon  its 
axis  towards  the  west  until  the  hour-index  points  to  this 
time. 

Note.  —  When  the  day-circle  divides  the  globe  at  the  poles,  every 
place  upon  a  meridian  may  be  brought  into  a  given  hour  of  the  day 
or  night ;  but  this  cannot  be  done  upon  parallels  lying  wholly  upon 
the  day-side  or  the  night-side. 


MOUNTING   AND    OPERATING    GLOBES.  6/ 

Exercises.  —  Bring  New  York  into  4-h.  2om.  p.  M. 

Ans.  —  New  York  is  brought  into  4h.  2om.  p.  M.  when 
its  meridian  is  4^  intervals  east  of  the  solar  index. 

Bring  Chicago  into  2  A.  M.,  Nashville  into  midnight, 
Mexico  into  i  A.  M.,  Lima  into  6h.  2om.  p.  M.,  London  into 
30.  4om.  A.  M.,  Constantinople  into  nh.  lorn.  A.  M. 


PROBLEM   XL 

Given  the  Difference  of  Longitude  between  two  Places ',  to  find 
their  Difference  of  Time. 

FIRST   METHOD. 

Reduce  the  difference  of  longitude  to  hours  by  dividing 
by  15,  or  to  minutes  by  multiplying  by  4. 

SECOND   METHOD. 

1.  Bring  the  solar  index  opposite  the  equator  by  bring- 
ing either  March  21  or  September  21  to  the  calendar  index. 

2.  Bring  the  meridian  of  Greenwich  to  the  solar  index. 

3.  Turn  the  globe  on  its  axis  towards  the  east  until  the 
solar  index  has  passed  over  as  many  degrees  as  are  equal 
to  the  given  difference  of  longitude.     The  number  of  in- 
tervals in  these  degrees  (if  there  are  15  degrees  in  an  inter- 
val) will  be  the  number  of  hours  in  the  required  difference. 

THIRD    METHOD. 

1.  Bring  the  meridian  of  Greenwich  to  the  solar  index. 

2.  Set  the  hour-index  to  XII. 

3.  Turn  the  globe  on  its  axis  towards  the  east  until  the 
difference  of  longitude  is  passed  by  the  solar  index.     The 
hour-index  will  now  point  to  the  required  difference  of 
time. 


68  NEW    METHOD    OF 

Exercises.  —  Find  the  difference  of  time  between  two 
places  whose  difference  of  longitude  is  40°. 

Ans.  —  2  h.  40  m. 

Find  the  difference  of  time,  the  difference  of  longitude 
being  10°,  124°,  50°,  67°,  90°,  84°. 


PROBLEM   XII. 

Given  the  Difference  of  Time  between  two  Places ',  to  find  their 
Difference  of  Longitude. 

FIRST    METHOD. 

Reduce  the  difference  of  time  to  longitude  by  multiply- 
ing it,  if  given  in  hours,  by  15  ;  or  dividing  it,  if  given  in 
minutes,  by  4. 

SECOND   METHOD. 

1.  Bring  the  solar  index  opposite  the  equator  by  bring- 
ing either  March  21  or  September  21  to  the  calendar  index. 

2.  Bring  the  meridian  of  Greenwich  to  the  solar  index. 

3.  Turn  the  globe  on  its  axis  either  towards  the  east  or 
the  west,  until  the  solar  index  has  passed  over  as  many 
intervals  of  longitude  as  are  equal  to  the  number  of  hours 
in  the  given  difference  of  time.     The  degree  of  longitude 
opposite  the  solar  index  is  the  required  difference  of  lon- 
gitude. 

THIRD   METHOD. 

1.  Bring  the  meridian  of  Greenwich  to  the  solar  index. 

2.  Set  the  hour-index  to  XII. 

3.  Turn  the  globe  on  its  axis  towards  the  east  until  the 
given  difference  of  time  is  passed  by  the  hour-index.     The 
degree  of  longitude  opposite  the  solar  index  is  the  required 
difference  of  longitude. 


MOUNTING   AND    OPERATING    GLOBES.  69 

Exercises.  —  Find  the  difference  of  longitude  between 
two  places  whose  difference  of  time  is  2  h.  20  m. 

Ans.  35°. 

Find  the  difference  of  longitude,  the  difference  of  time 
being  4  h.,  6  h.  30  m.,  10  h.,  16  h.,  3  h.  8  m.,  12  h.,  16  h. 


PROBLEM   XIII. 

Given  two  Places,  to  find  their  Difference  of  Time. 

FIRST   METHOD. 

Find  the  difference  of  longitude  between  the  two  places 
by  Problem  V.,  then  proceed  as  in  Problem  XI. 

SECOND   METHOD. 

1.  Bring  the  meridian  of  one  of  the  places  to  the  solar 
index,  having  the  meridian  of  the  other  upon  the  west 
of  it. 

2.  Turn  the  globe  on  its  axis  towards  the  east,  and  count 
the  intervals  of  longitude  that  are  passing  while  the  merid- 
ian of  the  second  place  is  brought  opposite  the  solar  index. 
This  number  is  the  number  of  hours  in  the  required  dif- 
ference. 

THIRD    METHOD. 

1.  Bring  the  meridian  of  one  of  the  places  to  the  solar 
index,  having  the  meridian  of  the   other  upon  the  west 
of  it. 

2.  Set  the  hour-index  to  XII. 

3.  Turn  the  globe  on  its  axis  towards  the  east  until  the 
meridian  of  the  second  place  is  brought  opposite  the  solar 
index.     The  hour-index  will  now  point  to  the  required  dif- 
ference. 

Exercises.  —  Find  the  difference  of  time  between  Savan- 
nah and  Sante  Fe. 
Ans.  —  i  h.  40  m. 


7O  NEW    METHOD    OF 

Find  the  difference  of  time  between  Alexandria  and 
Damascus,  Calais  and  Halifax,  Bangor  and  Smyrna,  Gal- 
veston  and  Ghent,  Hamburg  and  Havana,  Chicago  and 
Portland. 


PROBLEM   XIV. 

Given  the  Hour  of  the  Day  at  a  Place,  to  find  all  those  Places 
on  the  Earth 's  Surface  where  it  is  then  Noon,  or  any  other 
given  Hour. 

FIRST   METHOD. 

1.  Bring  the  given  place  into  the  given  hour  by  Problem 
X.     All  places  situated  upon  the  semi-meridian  opposite 
the  solar  index,  or  upon  the  one  which  bisects  the  day-side 
of  the  globe,  are  now  in  noon ;  all  places  situated  on  the 
semi-meridian  which  bisects  the  night-side  of  the  globe  are 
in  midnight. 

2.  If  the  hour  at  the  required  places  is  before  noon, 
the  meridian  upon  which  these  places  are  situated  will  be 
found  as  many  intervals  west  of  the  solar  index  as  there 
are  hours  before  noon. 

3.  If  the  hour  at  the  required  places  is  after  noon,  the 
meridian   upon  which  these  places  are  situated  will   be 
found  as  many  intervals  east  of  the  solar  index  as  there 
are  hours  after  noon. 

Note. — The  required  places  must  have  the  same  longitude;  that 
is,  they  must  be  found  upon  the  half  of  a  meridian  which  passes 
between  the  poles.  If  any  portion  of  this  semi-meridian  crosses 
parallels  lying  wholly  upon  the  day-side  or  the  night-side  of  the 
globe,  the  hour  at  places  upon  this  portion  is  the  same  as  at  places 
upon  the  remaining  portion  ;  but  this  hour  shows  the  advance  of  the 
sun  from  due  north  or  south  within  the  sky  (if  the  period  is  one  of 
continuous  day),  rather  than  the  advance  of  the  day  from  noon  or  mid- 
night. If  the  period  is  one  of  continuous  night,  the  advance  of  the 
sun  from  due  north  or  south,  below  the  horizon,  is  shown. 


MOUNTING   AND    OPERATING    GLOBES. 


SECOND    METHOD. 

1.  Find  the  difference  between  the  given  time  and  the 
time  at  the  required  places,  and  reduce  this  difference  to 
lengitude  by  Problem  XII. 

2.  If  the  hour  at  the  required  places  is  earlier  than  the 
hour  at  the  given  place,  the  required  meridian  will  be  found 
as  many  degrees  west  of  the  given  place  as  are  equal  to  the 
difference  of  longitude. 

3.  If  the  hour  at  the  required  places  is  later  than  the 
hour  at  the  given   place,   the   required  meridian  will  be 
found  as  many  degrees  east  of  the  given  place  as  are  equal 
to  the  difference  of  longitude. 

THIRD    METHOD. 

1.  Find  the  difference  between  the  hour  at  the  given 
place  and  the  hour  at  the  required  places. 

2.  Bring  the  meridian  of  the  given  place  to  the  solar 
index. 

3.  Set  the  hour-index  to  XII. 

4.  If  the  hour  at  the  required  places  is  earlier  than  the 
hour  at  the  given  place,  turn  the  globe  on  its  axis  towards 
the  east  until   the  hour-index  has  passed  over  as  many 
hours  as  are  equal  to  the  difference  of  time.     If  the  hour 
at  the  required  places  is  later  than  the  hour  at  the  given 
place,  turn  the  globe  on  its  axis  towards  the  west  until  the 
hour-index  has  passed  over  as  many  hours  as  are  equal  to 
the  difference  of  time ;   and   in  both  cases  the  required 
places  will  be  found  upon  the  meridian  opposite  the  solar 
index,  (a) 

(a)  The  use  made  of  the  solar  index  in  the  preceding  method  is 
convenient,  but  somewhat  objectionable,  because  it  places  a  merid- 
ian having  any  given  time  of  forenoon  or  afternoon  in  the  position 
of  noon  as  this  position  is  represented  upon  the  globe. 


72  NEW    METHOD    OF 

Exercises.  —  When  it  is  12  M.  at  Boston,  where  is  it 
7  A.M.  ? 

Ans.  —  At  Prince  Williams's  Sound,  and  at  the  central 
islands  of  King  George's  group  in  the  Pacific  Ocean. 

When  it  is  12  M.  at  London,  where  is  it  8  A.M.?  when 
5  P.M.  at  Madrid,  where  is  it  noon  ?  when  5h.  3om.  A.M.  at 
Pekin,  where  is  it  midnight  ?  when  3  A.M.  at  Delhi,  where 
is  it  6  A.M.  ?  when  noon  at  Boston,  where  is  it  i  A.M.  ? 
when  2  A.M.  at  Philadelphia,  where  is  it  2  P.M.  ? 


PROBLEM  XV. 

Given  the  Hour  of  the  Day  at  a  Place,  to  find  the  Hour 
at  another  Place,  (a) 

FIRST    METHOD. 

1.  Find  the  difference  of  time  between  the  two  places 
by  Problem  XIII. 

2.  Add  this  difference  to  the  given  time,  or  subtract  it 
from  it,  according  as  the  required  place  is  east  or  west  of 
the  given  place. 

SECOND    METHOD. 

1.  Find  the  difference  of   longitude  between  the  two 
places  by  Problem  V. 

2.  Reduce  this  difference  to  time  by  Problem  XL;  then 
proceed  according  to  direction  2  of  First  Method. 

THIRD    METHOD. 

Bring  the  given  place  into  the  given  hour  by  Problem 
X.  The  distance  of  the  meridian  of  the  other  place  from 
the  solar  index  will  now  indicate  the  required  hour. 

(a)  The  hour  is  the  same  at  places  whose  longitude  is  the  same. 


I 


MOUNTING   AND    OPERATING    GLOBES.  73 


Exercises.  —  Find  the  time  at  Rio  Janeiro  when  it  is 
noon  at  Madrid. 

Ans.  — gh.  28m.  A.M. 

Find  the  time  at  Philadelphia  when  it  is  8  A.M.  at  Lon- 
don, at  Paris  when  it  is  4  P.M.  at  Canton,  at  Boston  when 
it  is  midnight  at  Leghorn,  at  New  York  when  it  is  10  A.M. 
at  San  Francisco,  at  Havana  when  it  is  5  A.M.  at  Edin- 
burgh, at  Portland  when  it  is  2h^3om.  A.M.  at  Quebec. 


PROBLEMS   UPON  DAY  AND  NIGHT. 


PROBLEM  XVI. 

To  represent  the  Three  Positions  of  the  Sphere,  —  Right, 
Parallel,  and  Oblique. 

1.  Adjust  the  ring  arrangement  to  a  position  on  the 
equator  for  the  right  sphere. 

2.  Adjust  the  ring  arrangement  to  any  position  between 
the  equator  and  the  poles  for  the  oblique  sphere. 

3.  Adjust  the  ring  arrangement  to  the  North  Pole  for 
the  parallel  sphere. 

N.  B.  —  See  that  the  brass  meridian  is  always  set  in  the  direction 
of  a  meridian  of  the  globe.  Be  careful  also,  when  changing  the 
position  of  the  ring  arrangement,  to  lift  it  slightly,  that  it  may  rub 
upon  the  surface  of  the  globe  as  little  as  possible. 


74  NEW    METHOD    OF 


PROBLEM   XVII. 

To  find  the  Times  of  Sunrise  arid  Sunset  at  a  given  Place  on 
a  given  Day,  and  also  the  Lengths  of  the  Day  and  of  the 

Night,  (a) 

FIRST   METHOD. 

1.  Bring  the  given  day  to  the  calendar  index. 

2.  Look  for  the  parallel  upon  which  the  given  place  is 
situated,  and  count  the  intervals  of  longitude  upon  this 
parallel,  which  are  upon  the  day-side  of  the  globe,  for  the 
length  of  the  day. 

3.  Count  the  intervals  of  longitude  upon  the  same  par- 
allel, which  are  upon  the  night-side  of  the  globe,  for  the 
length  of  the  night ;  or  subtract  the  length  of  the  day  from 
24  hours. 

4.  One-half  the  length  of  the  day  is  the  time  of  sunset, 
and  one-half  the  length  of  the  night  is  the  time  of  sun- 
rise. 

SECOND   METHOD. 

1.  Bring  the  given  day  to  the  calendar  index. 

2.  Bring  the  meridian  of  the  given  place  to  the  solar 
index. 

3.  Set  the  hour-index  to  XII. 

4.  Turn  the  globe  on  its  axis  towards  the  west  until  the 
given  place  is  beneath  the  graduated  edge  of  the  day-circle, 
upon  the  left.     The  hour-index  will  now  point  to  the  time 
of  sunrise. 

5.  Turn  the  globe  on  its  axis  towards  the  east  until  the 
given  place  is  beneath  the  graduated  edge  of  the  day-circle, 
upon   the  right.     The  hour-index  will   now  point  to  the 
time   of  sunset. 

(a)  The  given  place  must  be  situated  upon  a  parallel  lying  partly 
upon  the  day-side  of  the  globe)d|and  partly  upon  the  night-side,  in 
order  to  have  the  6un  rise  and  set  every  day. 


MOUNTING   AND    OPERATING    GLOBES.  75 

6.  Twice  the  time  of  sunset  gives  the  length  of  the  day ; 
twice  the  time  of  sunrise  gives  the  length  of  the  night. 

Exercises.  —  Find  the  times  of  sunrise  and  sunset,  and 
the  lengths  of  the  day  and  the  night,  at  London,  upon 
July  17. 

Ans.  —  The  sun  rises  at  4  o'clock,  and  sets  at  8  :  length 
of  day,  1 6  hours;  length  of  night,  8  hours. 

Find  the  times  of  sunrise  and  sunset,  and  the  lengths  of 
the  day  and  the  night,  at  Washington,  May  i  ;  London, 
August  15  ;  New  Orleans,  July  10;  Mexico,  November  4; 
Cape  Horn,  February  5  ;  Quito,  May  i,  October  6,  De- 
cember 21.  (a) 

N.  B.  —  Whenever  it  is  not  convenient  to  find  the  length  of  day  or 
night  upon  a  parallel  in  the  southern  hemisphere,  find  this  length 
upon  the  parallel  corresponding  to  the  given  one  in  the  opposite  hemi- 
sphere, reversing  the  seasons  :  thus,  if  the  length  of  day  is  required 
upon  the  5oth  southern  parallel,  find  the  length  of  night  upon  the 
5Oth  northern  one,  and  vice  versa.  The  times  of  sunrise  and  sunset 
can  be  obtained  from  these  lengths,  according  to  direction  4  "of  First 
Method. 

Note,  —  If  the  length  of  a  number  of  successive  days  be  obtained 
at  places  differing  in  latitude,  this  length  will  be  found  to  change  at 
a  constantly  increasing  rate  as  we  go  towards  the  poles  :  thus  20° 
of  the  20th  parallel,  and  70°  of  the  5Oth,  pass  into  day  in  the  same 
interval  of  3  months.  Again  :  day  increases  from  o  to  12  hours  in  i 
month  and  20  days  upon  the  yoth  parallel,  and  in  18  days  upon  the 
8oth.  This  difference  in  the  rate  at  which  day  changes  in  length 
upon  different  parallels  is  due  to  the  varying  diameters  of  the  paral- 
lels :  the  smaller  the  diameter,  the  faster  the  rate  at  which  the  par- 
allel crosses  the  day-circle  :  thus  one-half  of  the  parallel  of  80°  N. 
latitude  crosses  this  circle  while  one-fourth  of  the  parallel  of  70°  N. 
latitude  is  crossing  it. 

(a)  The  times  of  sunrise  and  sunset,  length  of  day,  &c.,  as  found 
upon  a  globe,  vary  somewhat  from  astronomical  exactness.  This  is 
because  certain  causes,  which  give  rise  to  a  variable  rate  of  change  of 
the  sun's  right  ascension,  are  not  allowed  for  upon  a  globe.  For  an 
explanation  of  these  causes,  see  some  treatise  on  astronomy. 


76  NEW    METHOD    OF 


PROBLEM   XVIII. 

To  find  the  Length  of  a  Period  of  Continuous  Day  upon  a 
given  Parallel  in  the  North  Frigid  Zone;  also  when  this 
Period  begins  and  ends. 

1.  Bring  June  21  to  the  calendar  index. 

2.  Turn  the  calendar  disk  backward  until  the  given  par- 
allel has  its  farthest  point  from  the  solar  index  beneath 
the  day-circle,  and  note'  the  date  opposite  the  calendar 
index  for  the  beginning  of  the  given  period. 

3.  Turn  the  calendar  disk  forward  until  the  given  paral- 
lel has  its  farthest  point  from  the  solar  index  again  beneath 
the  day-circle,  and  note  the  date  opposite  the  calendar 
index  for  the  end  of  the  given  period. 

4.  Estimate  in  days  or  months,  upon  the  calendar  disk, 
the  distance  between  the  two  dates  for  the  length  of  the 
given  period,  (a) 

Exercises.  —  Find  the  beginning,  end,  and  duration  of  a 
period  of  continuous  day  upon  the  parallel  of  80°  N.  lati- 
tude. 

Arts.  —  Length  of  period,  four  months  :  it  begins  April 
21,  and  ends  August  21. 

Find  the  beginning,  end,  and  duration  of  a  period  of 
continuous  day  upon  the  parallels  of  70°,  75°,  and  80°  N. 
latitude. 


(a)  Problems  are  given  upon  parallels  in  the  frigid  zones,  instead 
of  places,  because  the  latter  are  so  few.  As  the  earth  rotates  upon 
its  axis,  a  place  follows  the  direction  of  its  parallel  as  regards  the 
sun.  Therefore,  so  long  as  either  a  part  or  the  whole  of  the  parallel 
is  in  day  or  night,  a  place  upon  it  has  a  day  or  night  which  is  either 
a  part  or  the  whole  of  24  hours. 


MOUNTING   AND    OPERATING   GLOBES.  77 


PROBLEM  XIX. 

To  find  the  Length  of  a  Period  of  Continuous  Night  upon  a 
given  Parallel  in  the  North  Frigid  Zone;  also  when  this 
Period  begins  and  ends. 

1.  Bring  December  21  to  the  calendar  index. 

2.  Turn  the  calendar  disk  backward  until  the  given  par- 
allel has  its  nearest  point  to  the  solar  index,  beneath  the 
day-circle  ;  and  note  the  date  opposite  the  calendar  index 
for  the  beginning  of  the  given  period. 

3.  Turn  the  calendar  disk  forward  until  the  given  paral- 
lel has  its  nearest  point  to  the  solar  index  again  beneath 
the  day-circle ;  and  note  the  date  opposite  the  calendar 
index  for  the  end  of  the  given  period. 

4.  Estimate  in  days  or  months,  upon  the  calendar  disk, 
the  distance  between  the  two  dates  for  the  length  of  the 
given  period.     , 

Exercises.  —  Find  the  beginning,  end,  and  duration  of 
a  period  of  continuous  night  upon  the  parallel  of  70°  N. 
latitude. 

Ans.  —  Length  of  period,  two  months  and  twelve  days : 
it  begins  November  15,  and  ends  January  27. 

Find  the  beginning,  end,  and  duration  of  a  period  of 
continuous  night  upon  the  parallels  of  75°,  80°,  and  85°  N. 
latitude. 

PROBLEM  XX. 

To  find  the  Length  of  a  Period  of  Alternate  Day  and  Night, 
succeeding  one  of  Continuous  Day,  upon  a  given  Parallel  in 
the  North  Frigid  Zone  ;  also  when  this  Period  begins  and 
ends. 

1.  Bring  September  21  to  the  calendar  index. 

2.  Turn  the  calendar  disk  backward  until  the  given  par- 
allel has  its  farthest  point  from  the  solar  index  beneath 


/8  NEW    METHOD    OF 

the  day-circle,  and  note   the  date  opposite   the  calendar 
index  for  the  beginning  of  the  given  period. 

3.  Turn  the  calendar  disk  forward  until  the  given  par- 
allel has  its  nearest  point  to  the  solar  index  beneath  the 
day-circle,  and  note  the  date  opposite  the  calendar  index 
for  the  end  of  the  given  period. 

4.  Estimate  in  days  or  months  upon  the  calendar  disk 
the  distance  between  the  two  dates  for  the  length  of  the 
given  period. 

Exercises.  —  Find  the  beginning,  end,  and  duration  of 
a  period  of  alternate  day  and  night,  succeeding  one  of 
continuous  day,  upon  the  parallel  of  80°  N.  latitude. 

Ans.  Length  of  period,  two  months  :  it  begins  August 
21,  and  ends  October  21. 

Find  the  beginning,  end,  and  duration  of  a  period  of 
alternate  day  and  night,  succeeding  one  of  continuous  day, 
upon  the  parallels  of  70°,  75°,  and  85°  N.  latitude. 


PROBLEM  XXI. 

To  find  the  Length  of  a  Period  of  Alternate  Day  and  Night, 
succeeding  one  of  Continuous  Night,  upon  a  given  Parallel  in 
the  North  Frigid  Zone ;  also  when  this  Period  begins  and 
ends. 

1.  Bring  March  21  to  the  calendar  index. 

2.  Turn  the  calendar  disk  backward  until  the  given  par- 
allel has  its  nearest  point  to  the  solar  index  beneath  the 
day-circle,  and  note  the  date  opposite  the  calendar  index 
for  the  beginning  of  the  given  period. 

3.  Turn  the  calendar  disk  forward  until  the  given  par- 
allel has  its  farthest  point  from  the  solar  index  beneath 
the  day-circle,  and  note  the  date  opposite   the  calendar 
index  for  the  end  of  the  given  period. 

4.  Estimate  in  days  or  months  upon  the  calendar  disk 


MOUNTING   AND    OPERATING    GLOBES.  79 

the  distance  between  the  two  dates  for  the  length  of  the 
given  period. 

Exercises.  —  Find  the  beginning,  end,  and  duration  of 
a  period  of  alternate  day  and  night,  succeeding  one  of 
continuous  night,  upon  the  parallel  of  80°  N.  latitude. 

Ans.  —  Length  of  period,  two  months:  it  begins  Febru- 
ary 21,  and  ends  April  21. 

Find  the  beginning,  end,  and  duration  of  a  period  of 
alternate  day  and  night,  succeeding  one  of  continuous  night, 
upon  the  parallels  of  70°,  75°,  and  85°  N.  latitude. 

N.  B.  —  Problems  upon  the  south  frigid  zone  are  worked  upon  the 
north  frigid,  reversing  the  seasons  of  day  and  night,  thus  :  — 

1.  Given  a  period  of  continuous  day  in.  the  south  frigid  zone,  solve 
the  problem  for  a  period  of  continuous  night  in  the  north  frigid. 

2.  Given  a  period  of  continuous  night  in  the  south  frigid  zone,  solve 
the  problem  for  a  period  of  continuous  day  in  the  north  frigid. 

3.  Given  a  period  of  alternate  day  and  night,  succeeding  one  of 
continuous  day,  in  the  south  frigid  zone,  solve  the  problem  for  a  period 
of  alternate  day  arid  night,  succeeding  one  of  continuous  night,  in  the 
north  frigid. 

4.  Given  a  period  of  alternate  day  and  night,  succeeding  one  of  con- 
tinuous night,  in  the  south  frigid  zone,  solve  the  problem  for  a  period 
of  alternate  day  and  night,  succeeding  one  of  continuous  day,  in  the 
north  frigid. 

Note.  —  Two  equal  periods  of  alternate  day  and  night,  and  two 
other  equal  periods,  —  namely,  one  of  continuous  day  and  one  of  con- 
tinuous night,  —  compose  the  year  in  the  frigid  zones  :  therefore  the 
length  of  a  period  of  alternate  day  and  night  subtracted  from  12 
months,  and  divided  by  2,  will  give  the  length  of  a  period  of  contin- 
uous day  or  of  continuous  night,  and  vice  versa.  The  length  of  these 
four  periods  is  determined  by  the  distance  of  the  given  parallel  from 
the  pole.  Thus  the  two  parallels  of  70°  and  80°  north  latitude  have 
each  a  period  of  continuous  day  or  of  continuous  night  while  the  in- 
terval between  the  parallel  and  the  polar  circle  is  carried  twice  across 
the  day-circle,  or  while  the  respective  arcs  of  5°  and  25°  cross  this 
circle.  Again  :  the  same  parallels  have  each  a  period  of  alternate  day 
and  night  while  the  diameter  of  the  parallel  crosses  the  day-circle,  or 
while  the  respective  arcs  of  40°  and  20°  cross  this  circle. 


8O  NEW    METHOD    OF 


PROBLEM  XXII. 

At  a  given  Place,  the  Day  being  given,  to  find  what  other 
Day  of  the  Year  is  of  the  same  Length. 

FIRST    METHOD. 

1.  Bring  the  given  day  to  the  calendar  index. 

2.  Note  the  degree  of  latitude  opposite  the  solar  index. 

3.  Turn  the  calendar  disk  forward  until  the  same  degree 
is  again  brought  opposite  the  solar  index. 

4.  Look   for  the   required   day  opposite   the   calendar 
index,  (a) 

SECOND    METHOD. 

Find  how  many  days  the  given  day  is  before  the  nearer 
solstice  ;  and  the  required  day  is  the  same  number  after  it, 
and  vice  versa. 

THIRD    METHOD. 

Find  how  many  days  the  given  day  is  before  the  nearer 
equinox ;  and  the  required  day  is  the  same  number  after 
the  subsequent  one,  and  vice  versa. 

Exercises.  —  What  day  of  the  year  is  of  the  same  length 
as  April  15  ? 

Ans.  — August  27. 

What  day  of  the  year  is  of  the  same  length  as  March  4, 
May  15,  August  30,  October  12,  December  21,  June  2  ? 

PROBLEM   XXIII. 

To  find  at  what  Latitude,  not  within  the  Polar  Circles,  a  given 
Day  is  of  a  given  Length. 

FIRST    METHOD. 

1.  Bring  the  given  day  to  the  calendar  index. 

2.  Bring  any  meridian  to  the  solar  index,  (b) 

(a)  The  degree  of  latitude  opposite  the  solar  index  shows  the 
sun's  declination.     This  is  discussed  in  a  problem  farther  on. 

(b)  The  broad  meridian  passing  through  XII.  in  the  hour-circle  is 
the  best  one  for  this  purpose. 


MOUNTING    AND    OPERATING    GLOBES.  8 1 

3.  Count  from  this  meridian  upon  the  equator,   either 
to  the  east  or  the  west,  as  many  intervals  of  longitude  as 
are  equal  to  half  the  hours  of  the  given  day. 

4.  The  same  meridian  being  opposite  the  solar  index  as 
at   first,  follow  along  the  meridian  reached  by  counting, 
either  towards  the  north  or  the  south,  according  as  the 
given  latitude  is  north  or  so'ith  of  the  equator,  until  this 
meridian  intersects  the  day-circle,  where  will  be  found  the 
required  latitude. 

SECOND   METHOD. 

1.  Bring  the  given  day  to  the  calendar  index. 

2.  Bring  the  meridian  of  Greenwich  to  the  solar  index. 

3.  Set  the  hour-index  to  XII. 

4.  Turn  the  globe  on  its  axis  towards  the  east  until  the 
index  has  passed  over  as  many  hours  as  are  equal  to  half 
the  length  of  the  given  day. 

5.  Keep  the  .globe  from  revolving  on  its  axis,  and  fol- 
low along  the  meridian  of  Greenwich  from  XII.  upon  the 
hour-circle   until  this    meridian    intersects   the  day-circle, 
where  will  be  found  the  required  latitude. 

Note.  —  When  the  required  latitude  is  south  of  the  equator,  it  may 
be  found  upon  the  north  by  substituting  the  length  of  the  night  of  the 
given  date  for  that  of  the  day.  Thus,  if  the  latter  is  16  hours,  solve 
the  problem  for  a  day  of  8  hours,  and  call  the  northern  latitude, 
which  is  found,  a  southern  one.  It  is  sometimes  more  convenient 
to  practise  this  method  upon  southern  parallels. 

Exercises.  —  In  what  degree  of  N.  latitude  is  the  length 
of  the  longest  day  16  hours  ? 

Ans.  —  In  latitude  49°. 

In  what  degree  of  north  latitude  is  day,  upon  the  2oth  of 
May,  1 6  hours  long?  night  upon  the  ist  of  December,  13 
hours  long  ?  In  what  degree  of  south  latitude  is  the  short- 
est day  8  hours  long  ?  In  what  degree  of  north  latitude 
does  the  sun  set  at  7  o'clock  on  the  loth  of  April?  In 
what  degree  of  north  latitude  is  the  longest  day  five  times 
the  length  of  the  night  ? 

6 


82  NEW    METHOD    OF 


PROBLEM   XXIV. 

At  a  given  Place,  the  Day  and  Hour  being  given,  to  find  how 
the  various  Regions  of  the  Earth  are  situated  with  regard 
to  Day,  Night,  Twilight,  6-r. 

1.  Bring  the  given  day  to  the  calendar  index. 

2.  Bring  the  given  place  into  the  given  hour  by  Problem  X. 
All  regions  upon  the  day-side  of  the  globe  are  now  in 

day ;  all  upon  the  night-side  are  in  night. 

It  is  noon  upon  the  semi-meridian  which  bisects  the  day- 
side  of  the  globe,  or  which  is  opposite  the  solar  index. 

It  is  midnight  upon  the  semi-meridian  which  bisects  the 
night-side  of  the  globe. 

The  sun  is  vertical  at  the  position  opposite  the  solar  index. 

It  is  sunrise  at  places  beneath  the  day-circle,  upon  the 
left  of  the  semi-meridian  which  is  in  noon. 

It  is  sunset  at  places  beneath  the  day-circle,  upon  the 
right  of  the  semi-meridian  which  is  in  noon. 

It  is  the  beginning  of  morning  twilight  at  places  beneath 
the  twilight-circle,  upon  the  west  of  the  semi-meridian  which 
is  in  noon  ;  those  parallels  being  excepted  which  lie  wholly 
upon  the  night-side  of  the  globe. 

It  is  the  end  of  evening  twilight  at  places  beneath  the 
twilight-circle,  upon  the  east  of  the  semi-meridian  which 
is  in  noon  ;  those  parallels  being  excepted  which  lie  wholly 
upon  the  night-side  of  the  globe. 

It  is  continuous  day  at  places  whose  parallels  lie  wholly 
upon  the  day-side  of  the  globe. 

It  is  continuous  night  at  places  whose  parallels  lie  wholly 
upon  the  night-side  of  the  globe. 

It  is  continuous  twilight  at  places  whose  parallels  lie 
wholly  within  the  section  of  twilight. 

It  is  continual  dark-night,  or  a  night  without  any  twi- 
light, at  places  whose  parallels  lie  upon  the  night-side  of 
the  globe,  beyond  the  section  of  twilight. 


MOUNTING   AND    OPERATING   GLOBES.  83 

Twilight  and  dark-night  alternate,  during  every  24  hours, 
at  places  whose  parallels  are  divided  by  the  twilight-circle, 
and  which  do  not  extend  upon  the  day-side  of  the  globe. 

Twilight  continues  from  sunset  to  sunrise  at  places 
whose  parallels,  being  divided  by  the  day-circle,  have  their 
unilluminated  arcs  wholly  within  the  section  of  twilight. 

Exercises. — When  it  is  4  h.  52  m.  A.M.  at  London  on 
March  5,  where  is  the  sun  rising,  setting,  &c.,  &c.  ? 

Ans.  —  The  sun  is  rising  at  Riga,  the  western  part  of 
Greece,  &c. ;  setting  at  the  Gulf  of  Anadir,  Fox  Islands, 
&c. ;  it  is  noon  at  Lake  Baikal,  Gulf  of  Tonquin,  &c. ; 
midnight  at  New  York,  Montreal,  &c. ;  morning  twilight  at 
Norway,  Sweden,  &c. ;  evening  twilight  at  Alaska,  the 
Sandwich  Islands,  &c. ;  day  at  China,  Australia,  &c. ; 
night  at  the  United  States,  Mexico,  &c. ;  the  sun  is  verti- 
cal at  Batavia. 

Where  is  the  sun  rising,  setting,  &c.,  &c.,  when  it  is  2  P.M. 
at  Washingtonj  October  10?  noon  at  Quebec,  May  i  ?  n 
A.M.  at  Dublin,  September  i  ?  4  P.M.  at  Rome,  Novem- 
ber 4?  6  h.  30  m.  A.M.  at  Constantinople,  December  10? 
ii  h.  15  m.  A.M.  at  Cairo,  April  16  ? 


PROBLEMS  UPON  THE  SUN. 

PROBLEM  XXV. 

To  pass  the  Globe  through  the  four  Seasons  of  Spring,  Sum- 
mer, Autumn,  and  Winter ;  also  to  show  how  the  Northern 
and  the  Southern  Hemispheres  are  illuminated  upon  a 
given  Day,  and  the  Sun's  Place  in  the  Ecliptic. 

i.  Pass  the  globe  through  a  northern  spring  and  south- 
ern autumn  by  turning  the  calendar  disk  forward  three 
months  from  March  21. 


84  NEW    METHOD    OF 

2.  Pass  the  globe  through  a  northern  summer  and  south- 
ern winter   by  turning   the   calendar  disk  forward  three 
months  from  June  21. 

3.  Pass  the  globe  through  a  northern  autumn  and  south- 
ern spring   by  turning   the  calendar  disk  forward  three 
months  from  September  21. 

4.  Pass  the  globe  through  a  northern  winter  and  south- 
ern summer  by  turning  the  calendar  disk  forward  three 
months  from  December  21. 

5.  Show  how  the  northern  and  the  southern  hemispheres 
are  illuminated  at  any  given  time  by  bringing  this  time 
to  the  calendar  index,  and  noting  the  most  northern  and 
southern  positions  of  the  day-circle;  also  the  position  at 
which  the  sun  is  vertical. 

6.  Look  for  the  sign  and  degree  opposite  the  given  day 
for  the  sun's  place  in  the  ecliptic. 

N.  B. — The  globe  being  set  in  position  for  a  given  time,  —  say 
August  24, —  what  this  time  is  designed  to  show  in  this  problem  may 
be  stated  as  follows  :  namely,  upon  August  24  the  sun  shines  from 
the  point  in  the  8oth  northern  parallel  the  farthest  from  it  to  the 
point  in  the  Both  southern  parallel  the  nearest  to  it ;  it  is  vertical  at 
the  loth  northern  parallel,  and  is  in  the  3d0  of  the  sign  Virgo  of  the 
ecliptic. 

Exercises.  —  Pass  the  globe  through  a  northern  winter, 
a  southern  autumn,  a  southern  spring,  a  northern  spring,  a 
northern  summer,  a  southern  winter. 

Show  and  state  how  the  northern  and  the  southern 
hemispheres  are  illuminated ;  also  the  sun's  place  in  the 
ecliptic  upon  July  10,  September  i,  November  18,  Decem- 
ber 21,  March  21,  June  i. 


MOUNTING  AND    OPERATING   GLOBES.  85 


PROBLEM  XXVI. 

To  show  the  Rate  at  which  the  Sun  moves  between  the  Trop- 
ics ;  a  Rate  bearing  a  close  Relation  to  the  Rate  at  which 
Day  or  Night  changes  in  Length,  or  the  Rate  at  which  the 
Sun 's  Diurnal  Arcs  increase  or  decrease  in  Extent. 

Turn  the  calendar  disk  either  from  June  21  to  Decem- 
ber 21,  or  from  December  21  to  June  21,  and  observe  the 
rate  at  which  the  solar  index  moves  towards  the  north  or 
the  south  upon  the  globe,  estimating  this  rate  in  degrees 
at  the  close  of  each  month. 

Note.  —  As  an  example  of  the  unequal  rate  at  which  day  increases 
or  decreases  from  month  to  month,  contrast  the  following  figures, 
showing  the  increase  of  day  at  Boston  from  December  21  to  June  21  : 
viz.,  29  m.  from  December  21  to  January  21 ;  73  m.  from  January  21 
to  February  21 ;  83  m.  from  February  21  to  March  21  ;  86  m.  from 
March  21  to  April  21  ;  71  m.  from  April  21  to  May  21  ;  and  30  m. 
from  May  21  to  June  21. 


PROBLEM  XXVII. 

The  Day  being  given,  to  find  the  Sun  'j  Longitude,  Right 
Ascension,  and  Declination. 

1.  Find  the    sun's   place   in    the  ecliptic   according  to 
direction  6,  Problem    XXV.  :   the  distance  of   this   place 
from  the   first   degree   of   Aries,   reckoning   towards  the 
right,  is  the  sun's  longitude. 

2.  Mark  where  this  longitude  occurs  upon  the  circle  of 
the  ecliptic,  as  drawn  upon  the  globe  ;  that  is,  pass  to  a 
point  as  many  degrees  east  of  the  first  point  of  Aries  as 
equal  the  sun's  longitude.     Follow  from  this  point  to  the 
equator   along   a   meridian,  and    the    distance   from   this 
second  point  to  the  first  degree  of  Aries  is  the  sun's  right 
ascension. 

3.  Bring  the  given  day  to  the  calendar  index,  and  the 


86  NEW    METHOD    OF 

degree  of  latitude  opposite  the  solar  index  is  the  sun's 
declination. 

Exercises.  —  Find  the  longitude,  right  ascension,  and 
declination  of  the  sun  on  May  10. 

Ans.  —  Longitude,  49°  ;  right  ascension,  47  °;  declina- 
tion, 8°. 

Find  the  longitude,  right  ascension,  and  declination  of 
the  sun  on  July  4,  November  6,  April  21,  June  i,  August 
10,  February  18. 

PROBLEM  XXVIII. 

The  Day  being  given,  to  find  those  Places  where  the  Sun  is 
Vertical,  or  in  the  Zenith,  at  Noon. 

1.  Find  the  sun's  declination  for  the  given  day  accord- 
ing to  direction  3,  Problem  XXVII. 

2.  Turn  the  globe  on  its  axis,  and  all  places  passing 
opposite  the  solar  index  have  the  sun  vertical  at  noon. 

Exercises.  —  Find  those  places  where  the  sun  is  vertical 
at  noon  upon  April  20. 

Ans.  —  It  is  vertical  at  places  upon  the  tenth  northern 
parallel ;  viz.,  at  Cochin,  Gulf  of  Siam,  Gulf  of  Darien, 
Trinidad  Island,  &c. 

Find  those  places  where  the  sun  is  vertical  at  noon 
upon  July  4,  June  21,  September  i,  November  15,  Janu- 
ary 8,  March  i. 

PROBLEM   XXIX. 

The  Day,  Hour,  and  Place  being  given,  to  find  where  the  Sun 
is  Vertical. 

1.  Bring  the  given  day  to  the  calendar  index. 

2.  Bring  the  given  place  into  the  given  hour  by  Prob- 
lem X. 

3.  Look  for  the  required  place  opposite  the  solar  index. 


MOUNTING   AND    OPERATING   GLOBES.  / 

Exercises.  —  When  it  is  n  P.M.  at  New  York  on  the  ist 
of  May,  where  is  the  sun  vertical  ? 

Ans.  —  At  Manilla. 

Where  is  the  sun  vertical  when  it  is  2  h.  20  m.  P.M.  at 
Madras  on  the  2yth  of  February?  8  h.  36  m.  A.M.  at  Lon- 
don on  March  28  ?  2  h.  15  m.  P.M.  at  Calcutta  on  April  24? 
ii  P.M.  at  Cape  Sable,  Florida,  on  June  21?  7  h.  40  m. 
A.M.  at  Cape  Verde  on  August  12?  9A.M.  at  Charleston 
on  October  7  ? 

PROBLEM   XXX. 

To  find  those  two  Days  of  the  Year  on  which  the  Sun  will  be 
Vertical  at  a  given  Place  in  the  Torrid  Zone. 

1.  Turn  the  calendar  disk  to  bring  the  latitude  of  the 
given  place  opposite  the  solar  index. 

2.  Note  the  date  opposite  the  calendar  index  for  one  of 
the  days. 

3.  Turn  the  calendar  disk  forward  until  the  latitude  of 
the  given  place  is  again  opposite  the  solar  index. 

4.  Note  the  date  opposite  the  calendar  index  for  the 
other  day. 

Exercises.  —  On  what  two  days  of  the  year  is  the  sun 
vertical  at  Batavia,  in  the  Island  of  Java? 

Ans.  —  On  the  4th  of  March  and  the  8th  of  October. 

On  what  two  days  of  the  year  is  the  sun  vertical  at 
Manilla  ?  Lima  ?  Gondar  ?  Singapore  ?  Cape  St.  Roque  ? 
Aden? 

PROBLEM   XXXI. 

The  Length  of  the  Day  at  any  Place  being  given,  to  find  the 
Sun  'j  Declination,  and  the  two  Days  of  the  Year  on  which 
it  has  this  Declination. 

FIRST    METHOD. 

i.  Bring  the  meridian  of  the  given  place  to  the  solar 
index. 


88  NEW    METHOD    OF 

2.  Turn  the  globe  on  its  axis,  either  towards  the  west  or 
the  east,  until  the  solar  index  has  passed  over  as  many 
intervals  of  longitude  as  are  equal  to  half  the  hours  of  the 
given  day. 

3.  Keep  the  meridian  which  is  now  opposite  the  solar 
index  in  this  position,  and  turn  the  calendar  disk  either 
forward  or  backward  until  the  given  place  is  beneath  the 
day-circle. 

4.  Look  for  one  of  the  required  days  opposite  the  calen- 
dar index. 

5.  Find  the  other  required  day  by  Problem  XXII. ;  or 
turn  the  calendar  disk  forward  (still  keeping  opposite  the 
solar  index  the  meridian  brought  to  it  by  turning  the  globe 
on  its  axis  the  required  amount)  until  the  given  place  is 
again  beneath   the  day-circle,  when  the  day  sought  will 
again  be  found  opposite  the  calendar  index. 

6.  Find  the  sun's  declination,  upon  both  days,  at  the 
degree  of  latitude  opposite  the  solar  index,  according  to 
direction  3,  Problem  XXVII. 

SECOND   METHOD. 

1.  Bring  the  meridian  of  the  given  place  to  the  solar 
index. 

2.  Set  the  hour-index  to  XII. 

3.  Turn  the  globe  on  its  axis  towards  the  east  until  the 
hour-index  has  passed  over  half  the  hours  of  the  given  day. 

4.  Keeping   the   meridian  which   is   now  opposite   the 
solar  index  in  this  position,  turn  the  calendar  disk  either 
forward  or  backward  until  the  given  place  is  beneath  the 
day-circle. 

5.  Look  for  one  of  the  required  days  opposite  the  calen- 
dar index. 

6.  Find  the  other  required  day  by  Problem  XXII. ;  or 
turn  the  calendar  disk  forward  (still  keeping  opposite  the 
solar  index  the  meridian  brought  to  it  by  turning  the  globe 
on  its  axis  the  required  amount)  until  the  given  place  is 


MOUNTING   AND    OPERATING    GLOBES.  89 

again  beneath  the  day-circle,  when  the  day  sought  will 
again  be  found  opposite  the  calendar  index. 

7.  Find  the  sun's  declination,  upon  both  days,  at  the 
degree  of  latitude  opposite  the  solar  index,  according  to 
direction  3,  Problem  XXVII. 

Exercises.  —  What  two  days  of  the  year  are  each  14 
hours  long  at  New  York  ?  and  what  is  the  sun's  declina- 
tion on  these  two  days  ? 

Ans.  —  The  6th  of  May  and  the  6th  of  August ;  sun's 
declination,  17°  N. 

What  two  days  of  the  year  are  each  10  hours  long  at 
Boston  ?  6  hours  long  at  Dublin  ?  13  hours  long  at  Rome  ? 
What  two  nights  of  the  year  are  each  7  hours  long  at  Cape 
Race  ?  1 8  hours  long  at  Sitka  ?  What  two  days  of  the 
year  are  each  14  hours  long  at  Sydney,  Australia?  Give 
the  sun's  declination  in  each  case. 

N.  B.  — This  last  problem  upon  Sydney  can  be  more  conveniently 
solved  upon  the  corresponding  place  in  the  northern  hemisphere,  or 
latitude  35°  N.,  longitude  same  as  that  of  Sydney.  If  this  is  done, 
the  length  of  the  night  of  the  required  dates,  or  the  difference  be- 
tween the  given  length  and  24  hours,  must  be  substituted  for  the 
given  length, — a  course  already  indicated  in  a  preceding  problem. 
(See  Problem  XXIIL,  Second  Method,  note.) 


PROBLEM   XXXII. 

A  Place  and  the  Day  of  the  Year  being  given^  to  find  how 
much  the  Sun  'j  Declination  must  vary  to  make  the  Day  a 
given  Time  longer  or  shorter  than  the  given  Day  ;  also  how 
many  Days  will  elapse  during  the  change. 

FIRST    METHOD. 

1.  Bring  the  given  day  to  the  calendar  index. 

2.  Note  the  sun's  declination. 

3.  Bring  the  meridian  of  the  given  place  to  the  solar 
index. 


90  NEW    METHOD    OF 

4.  Note,  upon  the  parallel  of  the  given  place,  the  longi- 
tude beneath  the  day-circle,  either  upon  the  east  or  the 
west. 

5.  Reduce  one-half  of  the  given  increase  or  decrease  of 
time   to   degrees   of    longitude   by   Problem   XII.,    First 
Method. 

6.  Keep  the  meridian  of   the  given  place  opposite  the 
solar  index,  and  turn  the  calendar  disk  forward  until  the 
number  of  degrees  of  longitude  obtained  have  passed  either 
into  day  or  night,  according  as   the  day  is  to  be  made 
longer  or  shorter  at  the  given  place. 

7.  Note  the  sun's  declination,   and   its  distance  in   de- 
grees from  the  declination  found  upon  the  given  day  will 
be  the  required  change  :  the  distance  of  the  calendar  index 
from  the  given  day  will  give  the  required  number  of  days. 

SECOND    METHOD. 

1.  Bring  the  given  day  to  the  calendar  index. 

2.  Note  the  sun's  declination. 

3.  Bring  the  meridian  of  the  given  place  to  the  solar 
index. 

4.  Set  the  index  to  XII. 

5.  Turn  the  globe  on  its  axis  towards  the  east  until  the 
given  place  is  beneath  the  day-circle. 

6.  Note  the  time  to  which  the  hour-index  points. 

7.  If  the  day  is  to  be  made  longer,  continue  to  turn  the 
globe   towards  the  east  until  the  hour-index  has  passed 
over  a  time  equal  to  one-half  of  the  given  increase  of  time. 

8.  If   the  day  is   to   be   made    shorter,  turn  the  globe 
towards  the  west  until  the  hour-index  has  passed  over  a 
time  equal  to  one-half  of  the  given  decrease  of  time. 

9.  Turn  the  calendar  disk  forward  (keeping  opposite  the 
solar  index  the  meridian  brought  to  it  by  turning  the  globe 
on  its  axis  the  required  amount)  until  the  given  place  is 
brought  beneath  the  day-circle. 


MOUNTING    AND    OPERATING   GLOBES.  9! 

10.  Note  the  sun's  declination,  and  its  distance  from  the 
declination  found  upon  the  given  day  will  be  the  required 
change  :  the  distance  of  the  calendar  index  from  the  given 
day  will  give  the  required  number  of  days. 

Exercises.  —  How  much  must  the  sun's  declination  vary, 
in  order  that  day  at  London  may  be  one  hour  longer  than 
on  the  29th  of  February  ?  also  how  many  days  will  elapse 
during  the  change  ? 

Ans.  On  the  2gth  of  February,  the  sun's  declination  is 
about  9°  S. ;  and  the  82d  degree  of  E.  longitude,  upon  the 
parallel  of  London  (the  meridian  of  London  being  opposite 
the  solar  index),  is  beneath  the  day-circle  upon  the  east. 
To  increase  the  day  an  hour,  7^°  must  be  added  to  the 
number,  or  the  Sgih  degree  of  E.  longitude  brought  to  the 
day-circle.  Turning  the  calendar  disk  forward  till  this  in- 
crease is  made,  the  meridian  of  London  being  constantly 
held  opposite  the-  solar  index,  the  sun  is  now  vertical 
at  the  fourth  southern  parallel,  and  the  calendar  index 
is  opposite  March  15  :  hence  the  sun's  declination  has  de- 
creased about  5°,  and  fifteen  days  have  elapsed,  while  day 
at  London  is  i  hour  longer  than  on  February  29. 

How  much  must  the  sun's  declination  vary,  that  day  at 
Boston  may  be  i  hour  shorter  than  on  the  3oth  of  June  ? 
that  night  at  Copenhagen  may  be  i  hour  longer  than  on 
January  i  ?  that  day  at  Canton  may  be  30  minutes  longer 
than  on  April  i  ?  that  day  at  Dublin  may  be  2  hours  longer 
than  on  May  i  ?  that  day  at  Stockholm  may  be  3  hours 
shorter  than  on  June  21?  that  day  at  Sydney,  Australia, 
may  be  i  hour  longer  than  on  July  i  ?  State  in  each  case 
how  many  days  will  elapse  during  the  change. 

N.  B.  —  The  problem  upon  Sydney,  Australia,  can  be  solved  upon 
the  parallel  of  35°  north  latitude,  instead  of  south,  if  more  conven- 
ient ;  but,  if  this  is  done,  day  must  decrease  instead  of  increase. 


92  NEW    METHOD    OF 


PROBLEM   XXXIII. 

To  find  in  what  Latitude  the  Sun  does  not  set  during  a  given 
Number  of  Days  in  the  Frigid  Zone. 

1.  If  the  required  latitude  is  north,  bring  June  21  to  the 
calendar  index;  if  south,  December  21.- 

2.  Turn  the  calendar  disk  forward  one-half  the  given 
number  of  days. 

3.  If  the  required  latitude  is  north,  find  it  at  the  northern 
parallel  whose  farthest  point  from  the  solar  index  is  be- 
neath the  day-circle. 

4.  If  the  required  latitude  is  south,  find  it  at  the  northern 
parallel  whose  nearest  point  to  the  solar  index  is  beneath 
the  day-circle,  this  parallel  being  upon  the  night-side  of  the 
globe. 

Exercises.  —  In  what  degree  of  north  latitude,  and  at 
what  places,  does  the  sun  continue  above  the  horizon  for 
78  days  ? 

Ans.  The  ;2d  degree,  or  at  Cape  North,  in  Lapland,  the 
southern  part  of  Nova  Zembla,  Point  Beechy,  &c.  In 
what  degree  of  north  latitude,  and  at  what  places,  does  the 
sun  continue  above  the  horizon  for  2  months  ?  4  months  ? 
20  days?  In  what  degree  of  south  latitude  does  the  sun 
continue  above  the  horizon  for  3  months  ?  5  months  ? 
50  days  ? 

PROBLEM   XXXIV. 

Given  a  Day,  to  find  the  Equation  of  Time,  or  the  Difference 
between  Sun -Time  and  Clock -Time. 

1.  Find  the  sun's  longitude  and  right  ascension  for  the 
given  day  by  Problem  XXVIII. 

2.  Subtract  the  smaller  from  the  greater,  and  reduce  the 
difference  to  minutes  by  multiplying  by  four.      If  the  Ion- 


MOUNTING   AND    OPERATING    GLOBES.  93 

gitude  is  the  smaller,  the  sun  is  slower  than  the  clock ;  if 
the  right  ascension  is  the  smaller,  the  clock  is  slower  than 
the  sun. 

Exercises.  —  What  is  the  difference  between  sun-time 
and  clock-time  on  the  i;th  of  July? 

Ans.  —  The  sun's  right  ascension  exceeds  its  longitude 
by  two  degrees:  hence  the  sun  is  eight  minutes  slower 
than  the  clock. 

What  is  the  equation  of  time  on  August  10  ?  June  21  ? 
March  i  ?  September  21  ?  November  4?  April  10? 


PROBLEM  XXXV. 

To  describe  the  Course  of  the  Sun,  with  respect  to  the  Horizon 
and  Visible  Heavens •,  at  a  given  Place  upon  a  given  Day. 

N.  B.  —  Remove  the  brass  circles,  and  place  the  ring  arrangement 
upon  the  globe.  Have  the  «w-notched  extremity  of  the  brass  merid- 
ian towards  you. 

1.  Bring  the  given  day  to  the  calendar  index. 

2.  Adjust  the  ring  arrangement  to  the  given  place,  (a) 

3.  Turn  the  globe  on  its  axis  to  bring  the  brass  horizon 
to  the  solar  index,  with  the  brass  meridian  upon  the  west 
of  it. 

4.  Note  the  degree  upon  the  brass  horizon  which  is 
opposite  the  solar  index  for  the  sun's  rising  amplitude. 
If  the  solar  index  is  opposite  the  equator,  the  sun  is  rising 
in  the  east ;  if  the  solar  index  is  north  or  south  of  the 
equator,  the  sun  is  rising  north  or  south  of  east. 

5.  Turn  the  globe  on  its  axis  towards  the  east  until  the 
brass  meridian  is  brought  opposite  the  solar  index. 

6.  Note  the  degree  upon  the  brass  meridian  which  is 
opposite  the  solar  index  for  the  sun's  meridian  altitude. 
Subtract  this  altitude  from  90°  for  the  sun's  zenith  dis- 

(a)  See  manner  of  doing  this  in  "Description  of  Globe,"  p.  55. 


94  NEW    METHOD    OF 

tance,  or  read  this  distance  between  the  solar  index  and 
the  perforation  which  is  at  the  given  place. 

7.  Continue  to  turn  the  globe  towards  the  east  until  the 
brass  horizon  is  again  brought  opposite  the  solar  index. 

8.  Note  the  degree  upon  the  brass  horizon  which  is  op- 
posite the  solar  index  for  the  sun's  setting  amplitude.     If 
the  solar  index  is  opposite  the  equator,  the  sun  is  setting 
in  the  west ;  if  the  solar  index  is  north  or  south  of  the 
equator,  the  sun  is  setting  north  or  south  of  west. 

9.  Continue  to  turn  the  globe  towards  the  east  until  the 
solar  index  is  a  third  time  brought  to  the  brass  horizon, 
thus  describing  the  night-arc  of  the  given  parallel. 

10.  Continue  to  turn  the  globe  towards  the  east  while 
the  required  course  of  the  sun  is  again  described,  and  con- 
trast the  length  of  the  day-arc  with  that  of  the  night-arc ; 
this  contrast  showing  that  of  day  and  night,  in  regard  to 
length,  at  the  given  place  upon  the  given  day. 

N.  B.  —  Whenever  the  sun's  course  is  to  be  described  at  a  place 
in  the  southern  hemisphere,  adjust  the  ring  arrangement  to  the  an- 
tipodes of  the  place  (see  Problem  VI.),  and  ascertain  the  sun's 
meridian  altitude  and  zenith  distance  upon  the  meridian  passing 
through  the  place,  the  brass  meridian  being  now  below  the  horizon. 
The  meridian  altitude  is  the  number  of  degrees  between  the  solar 
index  and  the  nearer  extremity  of  the  brass  meridian,  and  the  zenith 
distance  is  the  number  between  the  solar  index  and  the  given  place. 
When  the  solar  index  is  opposite  the  perforation  (in  the  northern 
hemisphere)  or  the  given  place  (in  the  southern),  the  sun  is  repre- 
sented in  the  zenith. 

When  adjusting  the  ring  arrangement  to  a  place  upon  the  equator, 
set  the  notched  extremity  of  the  brass  meridian  around  the  south 
pole. 

Exercises.  —  Describe  the  course  of  the  sun  at  London 
upon  December  21. 

Ans.  —  Upon  December  21,  at  London,  the  sun  rises 
42°  S.  of  E.,  attains  a  meridian  altitude  of  14°,  and  sets 
42°  S.  of  W. 

Describe  the  sun's  course  at  Spitzbergen   upon  March 


MOUNTING   AND    OPERATING    GLOBES.  95 

21  ;  at  Quito  upon  May  i  ;  at  Magadoxa  upon  November 
10 ;  at  Para  upon  November  i  ;  at  Tonquin  upon  Au- 
gust i  ;  at  Rome  upon  July  5  ;  at  Madrid  upon  April  15  ; 
at  Quebec  upon  October  7  ;  at  Cape  Prince  of  Wales  upon 
December  21  ;  at  Sydney,  Australia,  upon  June  21  ;  at 
Spitzbergen  upon  April  21  ;  at  Hammerfest,  Qualoe 
Islands,  upon  April  i  ;  at  the  North  Pole  upon  July  30  ; 
at  the  South  Pole  upon  January  5. 


PROBLEM  XXXVI. 

To  find  the  Sun  V  Altitude  and  Azimuth  at  a  given  Place, 
the  Day  and  Hour  being  known. 

1.  Bring  the  given  day  to  the  calendar  index. 

2.  Adjust  the  ring  arrangement  to  the  given  place. 

3.  Bring   the    given    place    into    the    given    hour  by 
Problem  X. 

4.  Attach  the  altitude  quadrant  to  the  perforation  in  the 
brass  meridian,  and  extend  it  to  the  brass  horizon  in  the  di- 
rection of  the  solar  index :  the  number  of  degrees  between 
this  index  and  the  brass  horizon  is  the  required  altitude  ; 
the  number  of  degrees  intercepted  on  the  brass  horizon 
from  the  north  or  the  south  point  is  the  required  azimuth. 

Exercises.  —  What  are  the  sun's  altitude  and  azimuth  at 
London  on  the  ist  of  May  at  10  A.M.? 

Ans.  —  Altitude,  47°  ;  azimuth,  136°  from  the  N.  point. 

Find  the  altitude  and  azimuth  of  the  sun  at  New  York, 
November  5,  2  P.M.;  Berlin,  June  10,  8  A.M.;  Rome, 
May  i,  4  P.M.;  Boston,  December  21,  i  P.M.,  3  P.M.; 
Spitzbergen,  July  21,  i  A.M.,  4  P.M.,  10  P.M. 


96  NEW    METHOD    OF 


PROBLEM  XXXVII. 

To  find  the  Region  of  the  Earth  in  which  a  Solar  Eclipse 
is  visible,  the  Time  of  its  Occurrence  at  Washington  being 
given. 

1.  Find  the  place  where  the  sun  is  vertical  at  the  begin- 
ning of  the  eclipse  by  Problem  XXIX. :  at  all  places  within 
35°  of  this  place,  the  beginning  of  the  eclipse  is  visible. 

2.  Repeat  the  process  for  the  time  of  the  end  of  the 
eclipse,  obtaining  a  region  where  the  end  of  the  eclipse 
is  visible.     At  those  places  which   are  common   to  both 
regions,  thus  found,  the  whole  eclipse  is  visible. 

N.B.  —  The  region  of  the  eclipse  may  be  defined  upon  the  globe 
by  moving  the  quadrant  around  the  place  illuminated  by  a  vertical 
sun,  and  observing  the  circle  swept  at  a  distance  of  35°. 

Exercises.  —  On  June  5,  1872,  there  was  an  annular 
eclipse  of  the  sun ;  beginning  at  7  h.  12  m.,  and  ending  at 
13  h.  10  m.,  Washington  time.  Where  was  this  eclipse 
visible  ? 

On  March  25,  1876,  there  was  an  annular  eclipse  of  the 
sun  ;  beginning  at  o  h.  21  m.,  and  ending  at  5  h.  32  m., 
Washington  time.  Where  was  this  eclipse  visible  ? 


PROBLEM   XXXVIII. 

To  find  the  Region  of  the  Earth  in  which  a  Lunar  Eclipse  is 
visible,  the  Time  of  its  Occurrence  at  Washington  being 
given. 

1.  Find  the  place  where  the  sun  is  vertical  at  the  begin- 
ning of  the  eclipse  by  Problem  XXIX.     At  all  places  upon 
the  night-side  of  the  globe,  the  beginning  of  the  eclipse  is 
visible. 

2.  Repeat  the  process  for  the  end  of  the  eclipse,  obtain- 
ing a  hemisphere  where  the  end  of  the  eclipse  is  visible. 


MOUNTING   AND    OPERATING    GLOBES.  97 

At  those  regions  which  are  common  to  both  hemispheres, 
thus  found,  the  whole  eclipse  is  visible. 

Exercises.  —  On  May  22,  1872,  there  was  a  partial  eclipse 
of  the  moon  ;  beginning  at  5  h.  32  m.,  and  ending  at  6  h. 
48  m.,  Washington  time.  Determine  whether  the  begin- 
ning and  end  of  this  eclipse  were  visible  at  Dublin. 

On  March  9,  1876,  there  was  a  partial  eclipse  of  the 
moon  ;  beginning  at  12  h.  13  m.,  and  ending  at  14  h.  13  m., 
Washington  time.  Where  was  this  eclipse  visible  ? 

Note.  — A  solar  eclipse  is  a  partial  or  total  obscuration  of  the  sun, 
caused  by  the  interposition  of  the  moon  between  the  earth  and  the 
sun. 

A  lunar  eclipse  is  a  partial  or  total  obscuration  of  the  moon, 
caused  by  the  interposition  of  the  earth  between  the  moon  and  the 
sun. 


PROBLEMS    UPON    TWILIGHT. 


PROBLEM   XXXIX. 

A  Place  and  Day  of  the  Year  being  given,  to  find  the  Length 
of  Morning  and  Evening  Twilight. 

Note.  —  A  place  having  morning  and  evening  twilight  must  be  sit- 
uated upon  a  parallel  that  extends  upon  both  sides  of  the  section  of 
twilight.  Morning  twilight  is  upon  the  west,  and  evening  twilight 
upon  the  east.  The  two  arcs  which  measure  these  periods  upon  a 
given  parallel  are  seen  to  contain  the  same  number  of  degrees,  since 
they  proceed  through  the  section  of  twilight  in  the  same  direction. 
Parallels  within  484°  of  the  equator  always  have  morning  and  even- 
ing twilight. 

FIRST    METHOD. 

1.  Bring  the  given  day  to  the  calendar  index. 

2.  Look  for  the  parallel  upon  which  the  given  place  is 

7 


98  NEW    METHOD    OF 

situated,  and  count  the  intervals  of  longitude  upon  this 
parallel  which  are  within  the  section  of  twilight,  either 
upon  the  east  or  the  west.  The  number  of  these  intervals 
is  the  number  of  hours  in  the  required  length,  (a) 

SECOND    METHOD. 

1.  Bring  the  given  day  to  the  calendar  index. 

2.  Set  the  hour-index  to  XII. 

3.  Bring  the  given  place   either   to   the   twilight-circle 
upon  the  west,  or  to  the  day-circle  upon  the  east. 

4.  Turn  the  globe  on  its  axis  towards  the  east  until  the 
given  place  is  carried  through  the  section  of  twilight.    The 
hour-index  will  now  point  to  the  required  length. 

Exercises.  —  Find  the  length  of  morning  and  evening 
twilight  at  London  on  the  23d  of  September. 

Ans.  —  Two  hours. 

Find  the  length  of  morning  and  evening  twilight  at  Bos- 
ton, June  21 ;  Havana,  October  10  ;  Santiago,  February  5  ; 
Dublin,  August  20 ;  Madrid,  November  i ;  Pekin,  Decem- 
ber 10. 


PROBLEM   XL. 

A  Place  and  Day  of  the  Year  being  given,  to  find  when  Morn- 
ing Twilight  begins,  and  Evening  Twilight  ends. 

FIRST   METHOD. 

1.  Bring  the  given  day  to  the  calendar  index. 

2.  Bring  the  meridian  of  the  given  place  to  the  solar 
index. 

3.  Count  the  intervals  of  longitude  which  lie  between 
this  meridian  and  the  twilight-circle  upon  the  west,  reck- 

(a)  These  intervals  can  be  more  readily  counted  upon  a  parallel 
by  bringing  the  oceanic  surface  of  the  globe  within  the  section  of 
twilight. 


MOUNTING   AND    OPERATING    GLOBES.  99 

oning  upon  the  parallel  of  the  given  place.  The  number 
obtained,  being  subtracted  from  12,  will  give  the  hour  when 
morning  twilight  begins. 

4.  Count  in  the  same  manner  towards  the  east.  The 
number  obtained  will  give  the  hour  when  evening  twilight 
ends ;  or  the  number  obtained  when  counting  towards  the 
west  will  also  give  it. 

SECOND    METHOD. 

Find  the  times  of  sunrise  and  sunset  by  Problem  XVII., 
and  the  length  of  twilight  by  Problem  XL.  This  length, 
subtracted  from  the  time ,  of  sunrise,  will  give  the  begin- 
ning of  morning  twilight ;  and  the  same  length  added  to 
the  time  of  sunset  will  give  the  end  of  evening  twilight. 

THIRD    METHOD. 

1.  Bring  the  given  day  to  the  calendar  index. 

2.  Bring  the  meridian  of  the  given  place  to  the  solar 
index. 

3.  Set  the  hour-index  to  XII. 

4.  Turn  the  globe  on  its  axis  towards  the  west  until  the 
given  place  is  brought  to  the  twilight-circle.     The  hour- 
index  will  now  point  out  the  beginning  of  morning  twi- 
light. 

5.  Turn  the  globe  on  its  axis  towards  the  east  until  the 
given  place  is  brought  to  the  twilight-circle  in  this  direc- 
tion.    The  hour-index  will  now  point  out  the  end  of  even- 
ing twilight. 

Exercises.  —  Find  when  morning  twilight  begins  and 
evening  twilight  ends  at  London  on  the  igth  of  April. 

Ans.  —  Morning  twilight  begins  at  2  h.  40  m.,  and  even- 
ing twilight  ends  at  9  h.  20  m. 

Find  when  morning  twilight  begins  and  evening  twilight 
ends  at  Washington,  April  10  ;  Cincinnati,  November  4  ; 
Cape  St.  Lucas,  January  15  ;  Jeddo,  August  i  ;  Manilla, 
October  21 ;  Valparaiso,  June  5. 


IOO  NEW    METHOD    OF 

N.  B.  —  Whenever  it  is  not  convenient  to  find  the  length  of  morn- 
ing and  evening  twilight  upon  a  southern  parallel,  find  this  length 
upon  the  corresponding  northern  parallel  for  a  date  six  months  for- 
ward. Thus,  if  the  given  day  at  the  southern  parallel  is  August  5, 
bring  February  5  to  the  calendar  index,  (a) 


PROBLEM   XLI. 

To  find  the  Length  of  a  Period  of  Twilight  lasting  from 
Sunset  to  Sunrise  at  a  given  Place  in  the  Temperate  Zone  ; 
also  when  this  Period  begins  and  ends. 

Note. — This  period  occurs  once  a  year  at  latitudes  higher  than 
48^°.  It  attains  the  middle  of  its  duration  on  June  21. 

1.  Bring  June  21  to  the  calendar  index. 

2.  Turn  the  calendar  disk  backward  until  the  parallel 
upon  which  the  given  place  is  situated  has  its  farthest 
point  from  the  solar  index,  beneath  the  twilight-circle  ; 
and  note  the  date  opposite  the  calendar  index  for  the 
beginning  of  the  given  period. 

3.  Turn  the   calendar  disk   forward   until   the  parallel 
upon  which  the  given  place  is  situated  again  has  its  far- 
thest point  from   the    solar   index,   beneath   the   twilight- 
circle  ;  and  note  the  date  opposite  the  calendar  index  for 
the  end  of  the  given  period. 

4.  Estimate  in  days  or  months,  upon  the  calendar  disk, 
the  distance  between  the  two  dates,  for  the  length  of  the 
given  period. 

Exercises.  —  Find  the  beginning,  end,  and  duration  of  a 
period  of  twilight  lasting  from  sunset  to  sunrise  at  Stock- 
holm. 

(o)  The  sun  moves  below  :.he  ho'rizon  of  a  place  at  the  same  time  and 
rate  that  the  place  moves  beyond  the  day-circle  into  night ;  and  hence 
it  is  that  twilight  is  as  correctly  measured  from  the  day-circle,  along 
the  parallel  of  the  given  place,  as  from  the  horizon,  along  the  circle 
of  declination  which  the  sun  is  describing. 


MOUNTING   AND    OPERATING    GLOBES.  IOI 

Ans.  —  Length  of  period,  3  months,  28  days :  it  begins 
April  22,  and  ends  August  20. 

Find  the  beginning,  end,  and  duration  of  a  period  of 
twilight  lasting  from  sunset  to  sunrise,  at  Dublin,  London, 
St.  Petersburg,  Tobolsk,  Okolsk,  Nain  (Labrador). 


PROBLEM   XLII. 

To  find  the  Length  of  a  Period  of  Twilight  lasting  from  Sun- 
set to  Sunrise  upon  a  given  Parallel  of  the  Frigid  Zones ; 
also  when  this  Period  begins  and  ends. 

Note.  —  This  period  occurs  twice  a  year  at  places  within  the  frigid 
zones. 

FIRST  METHOD. —  The  given  period  succeeding  one  of  con- 
tinuous day. 

1.  Bring  June  21  to  the  calendar  index. 

2.  Turn  the  calendar  disk  forward  until  the  given  par- 
allel has  its  farthest  point  from  the  solar  index,  beneath  the 
day-circle  ;  and  note  the  date  opposite  the  calendar  index 
for  the  beginning  of  the  given  period. 

3.  Continue  to  turn  the  calendar  disk  forward  until  the 
given  parallel  has  its  farthest  point  from  the  solar  index, 
beneath  the  twilight-circle ;  and  note  the  date  opposite  the 
calendar  index  for  the  end  of  the  given  period. 

4.  Estimate  in  days  or  months,  upon  the  calendar  disk, 
the  distance  between  the  two  dates,  for  the  length  of  the 
given  period. 

SECOND  METHOD.  —  The  given  period  succeeding  one  of 
continuous  night. 

1.  Bring  December  21  to  the  calendar  index. 

2.  Turn  the  calendar  disk  forward  until  the  given  par- 
allel has  its  farthest  point  from  the  solar  index,  beneath  the 


IO2  NEW    METHOD    OF 

twilight-circle  ;  and  note  the  date  opposite  the  calendar  in- 
dex for  the  beginning  of  the  given  period. 

3.  Continue  to  turn  the  calendar  disk  forward  until  the 
given  parallel  has  its  farthest  point  from  the  solar  index, 
beneath  the  day-circle ;  and  note  the  date  opposite  the  cal- 
endar index  for  the  end  of  the  given  period. 

4.  Estimate  in  days  or  months,  upon  the  calendar  disk, 
the  distance  between  the  two  dates,  for  the  length  of  the 
given  period. 

Exercises.  —  Find  the  beginning,  end,  and  duration  of  a 
period  of  twilight  lasting  from  sunset  to  sunrise,  and  suc- 
ceeding one  of  continuous  day,  upon  the  parallel  of  80°  N. 
latitude. 

Ans.  —  Length  of  period,  i  month,  18  days:  it  begins 
August  24,  and  ends  October  14. 

Find  the  beginning,  end,  and  duration  of  a  period  of 
twilight  lasting  from  sunset  to  sunrise,  and  succeeding  one 
of  continuous  day,  upon  the  parallels  of  70°  and  85°  N.  lati- 
tude, upon  the  Arctic  Circle. 

Find  the  beginning,  end,  and  duration  of  a  period  of 
twilight  lasting  from  sunset  to  sunrise,  and  succeeding  one 
of  continuous  night,  upon  the  parallels  of  70°  and  75°  N. 
latitude ;  upon  the  parallel  of  85°  S.  latitude. 

N.  B.  —  Whenever  the  length  of  a  period  of  twilight  upon  a 
southern  parallel  is  required  (except  when  twilight  occurs  as  morning 
and  evening),  obtain  this  length  upon  the  corresponding  parallel  in 
the  north  frigid  zone,  and  carry  the  dates  found  six  months  forward. 
Thus,  if  a  period  of  twilight  lasts  from  sunset  to  sunrise  from  August 
21  to  October  n  upon  the  8oth  northern  parallel,  it  lasts  from 
February  21  to  April  n  upon  the  Soth  southern  parallel. 


MOUNTING   AND    OPERATING    GLOBES.  1 03 


PROBLEM   XLIII. 

To  find  the  Length  of  a  Period  of  Continuous  Twilight  upon  a 
given  Parallel  in  the  Frigid  Zones;  also  when  this  Period 
begins  and  ends. 

Note.  —  This  period  occurs  twice  in  the  year  upon  parallels  within 
9°  of  the  poles. 

FIRST  METHOD.  —  The  given  period  beginning  a  period  of 
continuous  night. 

1.  Bring  December  21  to  the  calendar  index. 

2.  Turn  the  calendar  disk  backward  until  the  given  par- 
allel has  its  nearest  point  to  the  solar  index,  beneath  the 
day-circle ;  and  note  the  date  opposite  the  calendar  index 
for  the  beginning  of  the  given  period. 

3.  Turn  the  calendar  disk  forward  until   the  given  par- 
allel has  its  farthest  point  from  the  solar  index,  beneath  the 
twilight-circle  ;  and  note  the  date  opposite  the  calendar  in- 
dex for  the  date  of  the  given  period. 

4.  Estimate  in  days  or  months,  upon  the  calendar  disk, 
the  distance  between  the  two  dates,  for  the  length  of  the 
given  period. 

SECOND  METHOD.  —  The  given  period  ending  a  period  of 
continuous  night. 

1.  Bring  December  21  to  the  calendar  index. 

2.  Turn  the  calendar  disk  forward  until  the  given  par- 
allel has  its  farthest  point  from  the  solar  index,  beneath  the 
twilight-circle ;   and  note  the  date  opposite  the   calendar 
index  for  the  beginning  of  the  given  period. 

3.  Continue  to  turn  the  calendar  disk  forward  until  the 
given  parallel  has    its   nearest   point  to    the  solar  index, 
beneath  the  day-circle ;  and  note  the  date  opposite  the  cal- 
endar index  for  the  end  of  the  given  period. 

4.  Estimate  in  days  or  months,  upon  the  calemdar  disk, 


IO4  NEW    METHOD    OF 

the  distance  between  the  two  dates,  for  the  length  of  the 
given  period. 

Exercises.  —  Find  the  beginning,  end,  and  duration  of  a 
period  of  continuous  twilight  upon  the  parallel  of  85°  N. 
latitude  ;  this  period  beginning  one  of  continuous  night. 

Ans.  —  Length  of  period,  22  days:  it  begins  October  6, 
and  ends  October  28. 

Find  the  beginning,  end,  and  duration  of  a  period  of 
continuous  twilight  upon  the  parallels  of  82°  and  88°  N.  lati- 
tude ;  this  period  beginning  one  of  continuous  night. 

Find  the  beginning,  end,  and  duration  of  a  period  of 
continuous  twilight  upon  the  parallel  of  82°  N.  latitude  ;  at 
the  South  Pole,  this  period  ending  one  of  continuous  night. 


PROBLEM   XLIV. 

To  find  the  Length  of  a  Period  of  Twilight  alternating  every 
twenty-four  hours  with  Dark-Night  upon  a  given  Parallel 
in  the  Frigid  Zones ;  also  when  this  Period  begins  and  ends. 

Note. —  This  period  occurs  twice  in  the  year  upon  parallels  within 
5°  of  the  poles,  and  once  in  the  year  upon  larger  parallels. 

FIRST  METHOD.  —  The  given  parallel  being  at  a  distance  of 
of  more  than  9°  from  the  poles,  and  the  given  period  attend- 
ing the  entire  period  of  continuous  night. 

1.  Bring  December  21  to  the  calendar  index. 

2.  Turn  the  calendar  disk  backward  until  the  given  par- 
allel has  its  nearest  point  to  the  solar  index,  beneath  the 
day-circle ;   and  note  the  date  opposite  the  calendar  index 
for  the  beginning  of  the  given  period. 

3.  Turn  the  calendar  disk  forward  until  the  given  par- 
allel has  its  nearest  point  to  the  solar  index,  again  beneath 
the  day-circle ;  and  note  the  date  opposite  the  calendar  in- 
dex for  the  end  of  the  given  period. 


MOUNTING    AND    OPERATING    GLOBES.  10$ 

4.  Estimate  in  days  or  months,  upon  the  calendar  disk, 
the  distance  between  the  two  dates,  for  the  length  of  the 
given  period. 

SECOND  METHOD.  —  The  given  parallel  being  distant  from 
the  poles  less  than  9°  and  more  than  5°,  and  the  given  pe- 
riod occurring  between  the  periods  of  continuous  twilight. 

1.  Bring  December  21  to  the  calendar  index. 

2.  Turn  the  calendar  disk  backward  until  the  given  par- 
allel has  its  farthest  point  from  the  solar  index,  beneath  the 
twilight-circle ;  and   note  the  date  opposite  the  calendar 
index  for  the  beginning  of  the  given  period. 

3.  Turn  the  calendar  disk  forward  until  the  given  par- 
allel has  its  farthest  point  from  the  solar  index,  again  be- 
neath the  twilight-circle ;  and  note  the  date  opposite  the 
calendar  index  for  the  end  of  the  given  period.    • 

4.  Estimate  in  days  or  months,  upon  the  calendar  disk, 
the  distance  between  the  two  dates,  for  the  length  of  the 
given  period. 

THIRD  METHOD.  —  The  given  parallel  being  within  5°  of 
the  poles ,  and  the  given  period  preceding  one  of  continuous 
twilight. 

1.  Bring  December  21  to  the  calendar  index. 

2.  Turn  the  calendar  disk  forward  until  the  given  par- 
allel has  its  nearest  point  to  the  solar  index,  beneath  the 
twilight-circle  ;  and  note   the  date  opposite  the  calendar 
index  for  the  beginning  of  the  given  period. 

3.  Continue  to  turn  the  calendar  disk  forward  until  the 
given  parallel  has  its  farthest  point  from  the  solar  index, 
beneath  the  twilight-circle ;  and  note  the  date  opposite  the 
calendar  index  for  the  end  of  the  given  period. 

4.  Estimate  in  days  or  months,  upon  the  calendar  disk, 
the  distance  between  the  two  dates,  for  the  length  of  the 
given  period. 


IO6  NEW    METHOD    OF 

FOURTH  METHOD. •* — The  given  parallel  being  within  5°  of 
the  poles ;  and  the  given  period  following  one  of  continuous 
twilight. 

1.  Bring  December  21  to  the  calendar  index. 

2.  Turn  the  calendar  disk  backward  until  the  given  par- 
allel has  its  farthest  point  from  the  solar  index,  beneath  the 
twilight-circle ;  and  note  the  date  opposite   the   calendar 
index  for  the  beginning  of  the  given  period. 

3.  Turn  the  calendar  disk  forward  until  the  given  par- 
allel has  its  nearest  point  to  the  solar  index,  beneath  the 
twilight-circle ;  and  note  the  date  opposite  the  calendar 
index  for  the  end  of  the  given  period. 

4.  Estimate  in  days  or  months,  upon  the  calendar  disk, 
the  distance  between  the  two  dates,  for  the  length  of  the 
given  period. 

Exercises.  —  Find  the  beginning,  end,  and  duration  of  a 
period  of  twilight  alternating  every  24  hours  with  dark- 
night  upon  the  parallel  of  75°  N.  latitude. 

Ans.  —  Length  of  period,  3  months,  15  days:  it  begins 
October  30,  and  ends  February  15. 

Find,  according  to  First  Method,  the  length  of  a  period 
of  twilight  alternating  every  24  hours  with  dark-night 
upon  the  parallels  of  70°  and  77°  N.  latitude. 

Find,  according  to  Second  Method,  the  beginning,  end, 
and  duration  of  a  period  of  twilight  alternating  every  24 
hours  with  dark-night  upon  the  parallels  of  82°  and  84° 
N.  latitude. 

Find,  according  to  Third  Method,  the  beginning,  end, 
and  duration  of  a  period  of  twilight  alternating  every  24 
hours  with  dark-night  upon  the  parallels  of  86°  and  88° 
N.  latitude. 

Find,  according  to  Fourth  Method,  the  beginning,  end, 
and  duration  of  a  period  of  twilight  alternating  every  24 
hours  with  dark-night  upon  the  parallels  of  86°  and  88° 
N.  latitude. 


MOUNTING   AND    OPERATING   GLOBES.  IO/ 


PROBLEM   XLV. 

To  find  the  Length  of  a  Period  of  Dark-Night  upon  a  given 
Parallel  in  the  Frigid  Zones  ;  also  when  this  Period  begins 
and  ends. 
Note.  —  This  period  occurs  once  a  year  upon  parallels  within  5°  of 

the  poles. 

1.  Bring  December  21  to  the  calendar  index. 

2.  Turn  the  calendar  disk  backward  until  the  given  par- 
allel has  its  nearest  point  to  the  solar  index,  beneath  the 
twilight-circle ;  and  note  the  date  opposite  the  calendar 
index  for  the  beginning  of  the  given  period. 

3.  Turn  the  calendar  disk  forward  until  the  given  par- 
allel has  its  nearest  point  to  the  solar  index  again  beneath 
the  twilight-circle,  and  note  the  date  opposite  the  calendar 
index  for  the  end  of  the  given  period. 

4.  Estimate  in  days  or  months,  upon  the  calendar  disk, 
the  distance  between  the  two  dates,  for  the  length  of  the 
given  period. 

Exercises.  —  Find  the  beginning,  end,  and  duration  of  a 
period  of  dark-night  upon  the  parallel  of  88°  N.  latitude. 

Ans. —  Length  of  period,  i  month,  15  days:  it  begins 
November  27,  and  ends  January  12. 

Find  the  beginning,  end,  and  duration  of  a  period  of 
dark-night  upon  the  parallel  of  86°  N.  latitude  ;  at  the 
North  Pole ;  at  the  South  Pole. 

N.  B.  —  Whenever  the  length  of  a  period  of  dark-night  upon  a 
southern  parallel  is  required,  obtain  this  length  upon  the  correspond- 
ing northern  parallel,  and  carry  the  dates  found  six  months  forward, 
as  was  stated  concerning  twilight  at  the  close  of  Problem  XLII. 


IO8  NEW    METHOD    OF 


MISCELLANEOUS    PROBLEMS. 


1.  FIND  the  difference  of  longitude  between  New  York  and 
San  Francisco. 

2.  At  what  rate  per  hour  are  the  inhabitants  of  Botany  Bay 
carried  from  west  to  east  by  the  rotation  of  the  earth  on  its 
axis  ? 

3.  Bring  Washington  into  9  o'clock  A.M. 

4.  Find  the  difference  of  time  between  Boston  and  Rome. 

5.  When  it  is  2  o'clock  P.M.  at  Cincinnati,  where  is  it  n  P.M.  ? 

6.  Find  the  times  of  sunrise  and  sunset,  and  the  lengths  of 
the  day  and  the  night,  at  Paris  upon  May  14. 

7.  What  is  the  length  of  a  period  of  continuous  day  at  the 
northern  extremity  of  Nova  Zembla  ?  and  when  does  this  period 
begin  and  end  ? 

8.  Find  the  beginning,  end,  and  duration  of  a  period  of  alter- 
nate day  and  night,  succeeding  one  of  continuous  day,  upon  the 
parallel  of  75°  S.  latitude. 

9.  In  what  degree  of   south  latitude  is  the  longest  day  16 
hours  in  length  ? 

10.  What  two  days  of  the  year  are  each   17  hours  long  at 
Stockholm  ? 

11.  In  what  latitude  north  is  the  longest  day  eleven  times  the 
length  of  the  shortest  ? 

12.  Find  the  longitude,  right  ascension,  and  declination  of  the 
sun  on  March  10. 

13.  Where  is  the  sun  vertical  when  it  is   I  h.  40  m.  P.M.  at 
Edinburgh  ? 

14.  On  what  two  days  of  the  year  is  the  sun  vertical  at  Bom- 
bay ? 

15.  How  much  must  the  sun's  declination  vary,  that  day  at 
Liverpool  may  increase  two  hours  from  December  21  ?  and  how 
many  days  will  elapse  during  the  change  ? 

1 6.  In  what  degree  of  north  latitude,  and  at  what  places,  does 
the  sun  continue  above  the  horizon  for  40  days  ? 


MOUNTING    AND    OPERATING    GLOBES.  lOQ 

17.  What  is  the  equation  of  time  on  November  i  ? 

1 8.  Describe  the  course  of  the  sun  at  Paris  on  June  10. 

19.  Find  the  altitude  and  azimuth  of  the  sun  at  New  York  on 
March  i,  4  o'clock  P.M. 

20.  Find  the  length  of  morning  and  evening  twilight  at  Vienna 
on  September  i. 

21.  On  what  day  of  the  year  is  twilight  eight  hours  long  at 
London  ? 

22.  Find  the  beginning,  end,  and  duration  of  a  period  of  twi- 
light, lasting  from  sunset  to  sunrise,  at  the  Orkney  Islands. 

23.  When  the  sun's  meridian  altitude  at  London  is  30°,  what 
day  of  the  year  is  it  ? 

24.  On  what  day  of  the  year  is  the  sun's  meridian  altitude  at 
Paris  equal  to  the  latitude  of  Paris  ? 

25.  What  is  the  sun's  altitude  at  Philadelphia  on  May  7, 
3  o'clock  P.M.  ? 

26.  What  is  the  sun's  greatest  meridian  altitude  at  the  south- 
ern extremity  of  Patagonia  ? 

27.  In  what  latitude  north  does  the  sun  begin  to  shine  con- 
stantly on  April  10  ? 

28.  If  the  meridian  altitude  of  the  sun  on  June  7  is  50°,  what 
is  the  latitude  of  the  place  ? 

29.  At  what  hour  does  the  sun  rise  at  London  when  it  sets  at 
7  o'clock  at  St.  Petersburg  ? 

30.  What  place  of  the  earth  has  the  sun  in  the  zenith  when 
it  is  7  o'clock  A.M.  at  London  on  April  25  ? 

31.  Find,  in  the  southern  hemisphere,  a  city  at  which  the 
meridian  altitude  of  the  sun  on  December  21  is  79°  30',  the  sun 
being  towards  the  northern  point  of  the  horizon,  and  the  chro- 
nometer showing  London  time  to  be  i  h.  56  m.  A.M.  when  it  is 
noon  in  the  city. 

32.  Find  a  city  in  which  one  has  no  shadow  on  the  longest 
day  of  the  year,  and  on  every  other  day  the  shadow  falls  towards 
the  north,  and  at  which  it  is  noon  when  it  is  8  h.  12  m.  A.M.  at 
London. 

33.  Find  a  cape  in  the  northern  hemisphere  at  which,  on  the 
longest  day  in  the  year,  the  sun  sets,  and,  without  any  inter- 
vening night,  rises  immediately  in  the  same  part  of  the  hori- 
zon ;  and  also  where  it  is  12  h.  40  m.  A.M.  when  it  is  noon  at 
London. 


HO  NEW    METHOD    OF 


QUESTIONS   FOR   EXAMINATION. 


SECTION    I. 

1.  DEFINE  a  point,  line,  straight  line,  curve,  surface,  plan& 
curved  surface. 

2.  Define  a  circle  and  its  parts.     Point  out  these  parts  in 
Fig.  i. 

3.  Define  an  angle.     How  are  angles  measured  ?     How  is 
the  circumference  of  a  circle  divided  ?     To  what  is  the  absolute 
length  of  a  degree  proportional  ?     What  may  we  take  as  the 
measure  of  an  angle  ?    Define  a  right  angle  ;  angular  distances  ; 
a  diedral  angle.     State  the  measure  of  the  angles  represented  in 
Figs.  2  and  3. 

4.  Define  a  solid,  a  sphere  and  its  parts.     Define  and  illustrate 
great  and  small  circles.     Define  and  illustrate  the  poles  of  a 
great  circle  of  a  sphere. 

5.  What  is  said  regarding  angular  motion  ?  angular  velocity  ? 

SECTION    II. 

6.  Describe  the  shape  of  the  earth.     What  is  its  greatest  cir- 
cumference ?  its  greatest  diameter  ?  its  smallest  diameter  ? 

7.  What  is  a  terrestrial  globe  ? 

8.  Define  the  axis  of  the  earth,  its  poles,  equator,  meridians, 
parallels  of  latitude,  the  tropics,  polar  circles,  zones. 

9.  What  is  said  of  latitude  ?  what  of  longitude  ? 

10.  Prove   that  any  two   meridians   include   between   them, 
whether  measured  upon  the  equator  or  a  parallel,  the  same 
number  of  degrees. 

11.  What  does  Fig.  5  illustrate? 


MOUNTING   AND    OPERATING   GLOBES.  1 1  I 


SECTION    III. 

12.  Define  the  celestial  sphere,  poles,  equator,  celestial  me- 
ridian of  a  place,  parallels  of  declination,  hour-circles. 

13.  Define  the  horizon.     Distinguish  between  the  rational  and 
the  sensible  horizon.     Define  the  zenith,  nadir,  vertical  circles, 
the  prime  vertical,  the  cardinal  points  of  the  horizon. 

14.  Define  altitude,  meridian  altitude,  zenith  distance,  azimuth, 
amplitude,  declination,   right  ascension,  right  sphere,  parallel 
sphere,  oblique  sphere. 

15.  Define  the  ecliptic,  and  state  its  divisions.     What  are  the 
cardinal  signs  of  the  ecliptic  ?     What  is  said  concerning  the 
cardinal  points  of  the  ecliptic  ? 


SECTION    IV. 

N.  B.  —  The  numbers  following  questions  in  this  section  indicate  the 
article  in  which  the  answer  is  to  be  found.  Answers  continue  to  be  found 
in  the  same  article  until  a  new  number  occurs. 

16.  What  two  motions  of  the  heavenly  bodies  are  apparent 
in  the  sky?     56. 

17.  What  question  arises  regarding  these  two  motions  ?     57. 

18.  Is  it  easy  for  an  inhabitant  of  the  earth  to  conceive  it  in 
motion  ?     58.     State  cases  in  which  it  is  difficult  to  distinguish 
between  a  state  of  motion  and  one  of  rest. 

19.  Granting  that  the  movements  of  the  heavenly  bodies  may 
belong  either  to  themselves  or  to  the  earth,  which  is  the  more 
probable  supposition  ?     59. 

20.  Mention  experiments  furnishing  proofs  of  the  earth's  rota- 
tion.    60. 

21.  How  may  the  annual  movement  of  the  heavenly  bodies  be 
made  evident  to  the  eye  ?     61.     What  renders  it  probable  that 
the  earth  has  an  annual  motion  around  the  sun  ? 

22.  In  what  common  plane  do  the  earth  and  sun  at  all  times 
lie  ?     62.      What  is  the  shape  of  the  orbit  of  the  earth  within 
this  plane  ?     Show  wherein  an  ellipse  differs  from  a  circle. 

23.  Does  the  ellipse  of  Fig.  9  represent  correctly  the  ellipse 
described  by  the  earth  round  the  sun  ? 


112  NEW    METHOD    OF 

24.  When  is  the  earth  said  to  be  in   perihelion?   when  in 
aphelion  ?     What  is  its  distance  from  the  sun  in  these  two  posi- 
tions ? 

25.  What  is  the  extent  of  the  earth's  orbit  ?  the  velocity  of 
the  earth  in  this  orbit  ? 

26.  Describe  the  direction  which  the  earth  takes  in  its  orbit. 
63.     How  does  this  direction  compare  with  that  of  the  sun  in 
the  ecliptic  ?.    Give  the  illustration. 

27.  How  is  the  axis  of  the  earth  directed  regarding  the  plane 
of  its  orbit  ?     64. 

28.  How  is  it  shown  that  the  planes  of  the  ecliptic  and  the 
equator  are  inclined  to  each  other  ? 

29.  Give  the  illustration  of  the  pencil,  explaining  the  paral- 
lelism of  the  earth's  axis.     65. 

30.  Show  why  the  earth's  axis  always  appears  to  be  directed 
towards  the  same  points  in  the  heavens. 

31.  What  causes  the  alternation  of  day  and  night  ?   66.    Show 
why  it  is  that  the  sun  illuminates  one-half  of  the  earth's  surface 
at  all  times.     What  is  meant  by  the  day-circle  ? 

32.  .How  may  we  illustrate  the  alternation  of  day  and  night  ? 

33.  When  is  there  a  day  and  a  night  of  12  hours  each  over 
the  entire  earth  ?     67.     Give  facts  showing  the  variations  which 
these  undergo  in  length  in  the  course  of  the  year. 

34.  What  attendant  changes  take  place  in  the  diurnal  course 
of  the  sun  ?     68.     To  what  common  cause  are  these  changes  and 
those  of  the  length  of  the  day  due  ? 

35.  What  are  shown  in  Fig.  12  ?     69. 

36.  What  is  said  of  the  positions  marked  vernal  and  autumnal 
equinox  ?     Why  are  day  and  night  always  12  hours  in  length  at 
the  equator  ? 

37.  Why  are  March  21   and  September  21  called  the  equi- 
noxes ?     Date  the  equinoxes  in  the  southern  hemisphere. 

38.  How  far  from  the  equator  do  the  sun's  rays  extend  upon 
21  and  September  21  ?     Why  are  these  two  dates  called  the  sol- 
stices ?     Date  the  solstices  in  the  two  hemispheres. 

39.  Astronomically  considered,  when  do  the  four  seasons  of 
the  year  begin  ? 

40.  Follow  out,  with  the  aid  of  Fig.   12,  the  change  of  day 
and  night  between  these  several  dates.     70. 

41.  What   movement   of   the   northern  and   southern  hemi- 


MOUNTING   AND    OPERATING   GLOBES.  113 

spheres,  regarding  the  sun;  do  these  changes  show  to  us  ?    Illus- 
trate this  by  reference  to  Fig.  n. 

42.  How  does  approach  towards  the  pole  affect  the  difference 
in  the  length  of  day  and  night  ? 

43.  Why  are  the  tropics  and  polar  circles  distinguished  from 
the  other  parallels  upon  the  earth's  surface  ?     71. 

44.  When  do  parallels  have  a  day  and  a  night  every  twenty-four 
hours  ?     72.     When  does  their  day  or  night  continue  longer  than 
24  hours  ?     Why  is  it  that  the  parallels  of  the  frigid  zones  are 
subject  to  a  greater  variety  of  day  and  night,  as  regards  length, 
than  occurs  within  the  temperate  and  torrid  zones  ? 

45.  Name  the  four  periods  of  day  and  night  which  these  par- 
allels may  be  said  to  have  during  the  year. 

46.  State  the  middle  of  these  four  periods. 

47.  Give  an  instance  of  this  variety  of  day  and  night  within 
the  frigid  zones.     73. 

48.  What  is  the  length  of  continuous  day  and  night  in  the 
southern  part  of  Nova  Zembla  ?  of  the  periods  of  alternate  day 
and  night  ?    What  change  goes  on  in  these  latter  periods  with 
removal  towards  the  poles  ? 

49.  What  is  the  greatest  length  of  day  within  the  torrid  zone  ? 
within  the  temperate  zones  ?     How  may  the  length  of  the  night 
be  obtained  from  that  of  the  day,  and  vice  versa? 

50.  What  is  said  of  the   changes  of  day  and  night,  as  to 
length,  at  the  same  place  ?     74.     Explain  why  this  change  pro- 
ceeds the  slowest  at  the  times  of  the  solstices,  and  the  fastest  at 
the  times  of  the  equinoxes.     Estimate  the  rate  at  which  the 
sun's  declination  increases  from  the  time  of  the  vernal  equinox. 
How  does   the  sun's  variable  rate  of   movement  between  the 
tropics  affect  its  diurnal  course  at  any  given  place  ? 

51.  Why  and  when  do  we  have  days  of  the  year  equalling 
each  other  in  length  ?     75. 

52.  What  is  the  sum  total  of  daylight  at  every  place  upon  the 
earth's  surface  ?     76.     How  is  this  sum  made  up  at  the  poles  ? 
at  the  equator  ?  between  these  positions  ? 

53.  Were  the  earth's  axis  either,  perpendicular  to  the  plane  of 
its  orbit,  or  coincident  with  it,  what  would  follow  regarding  the 
length  of  day  and  night  over  the  earth's  surface  ?     Mention  a 
condition  which  would  bring  about  a  constant  length  of  day  and 
night  at  places  in  the  same  latitude,  while  this  length  would  vary 

8 


114  NEW    METHOD    OF 

at  places  differing  in  latitude.  What  condition  would  bring 
about  a  greater  variety  than  we  at  present  have  ?  what  a  less 
variety  ? 

THE   DIURNAL   COURSE   OF   THE   SUN. 

54.  In  what  way  may  we  continue  a  study  of  the  changes 
which  take  place  in  the  length  of  day  and  night  ?    77. 

55.  At  what  rate  does  the  sun  move   above  and   below  the 
horizon  ?     What  kind  of  a  circle  does  the  sun  describe  upon 
March  21  and  September  21?    at  other  times?     Describe  its 
appearance  at  the  north  pole  upon  March  21  and  September  21  ; 
during  the  three   subsequent   months.      Is   the  sun's  diurnal 
course  an  exact  circle  ? 

56.  Estimate  the  sun's  advance  above  a  polar  horizon,  and  re- 
turn to  it,  during  a  period  of  continuous  day.     With  what  is  the 
motion  of  the  sun  from  and  to  a  polar  horizon  identical  ? 

57.  What  is  said  of  the  points  of  the  compass  at  the  poles  ? 

58.  What  portion  of  its  diurnal  circle  does  the  sun  always 
describe   above  the   horizon   at   the  equator  ?    78.      Give   the 
reason  for  the  sun's  rising  in  the  east  and  setting  in  the  west 
at  all  places  upon  the  earth's  surface  on  March  21  and  Septem- 
ber 21. 

59.  Describe  the  sun's  course  at  the  equator  during  the  year. 

60.  When  does  the  horizon  of  a  place  between  the  equator  and 
the  poles  have  one-half  of  the  sun's  diurnal  course  described 
above  it  ?    79.    When  more  than  one-half  of  this  course  ?  when 
less? 

61.  Why  does  the  sun  rise  and  set  at  a  more  rapid  advance 
along  the  horizon  as  we  remove  from  the  equator  ? 

62.  What  places  have  one  diurnal  circle  described  above  their 
horizons  during  the  year  ? 

63.  Describe  the  manner  in  which  the  rising  and  the  setting 
sun  advances   along  the  horizons   of  places  within   the  polar 
circles.     80.     In  what  direction  does  the  angle  between  these 
circles  and  the  horizon  diminish  ? 

64.  Upon  what  does  the  rate  at  which  the  sun's  diurnal  circles 
ascend  above  the  horizon,  or  descend  towards  it,  depend  ? 

65.  Follow  the  sun's  course  at  Spitzbergen  during  a  year.    81. 

66.  How  may  the  distance  at  which  the  sun  culminates  from 


MOUNTING    'AND    OPERATING   GLOBES.  115 

the  zenith  be  known,  the  sun's  declination   being  given  ?     82. 
Give  examples. 

.67.  What  is  the  sun's  meridian  altitude  at  Boston,  its  declina- 
tion being  10°  N.  ?  at  New  Orleans,  its  declination  being  14°  S.  ? 

68.  What  is  said  of  the  sun's  culmination  within  and  beyond 
the  torrid  zone.     83. 

69.  What  is  said  of  the  sun's  culmination  at  the  times  of  the 
solstices  ? 

MEASUREMENT   OF   TIME. 

70.  Mention  different  methods  employed  to  measure  time.   84. 
What  is  the  great  standard  of  time  ? 

71.  Define  the  transit  of  a  heavenly  body.     85. 

72.  What  is  a  sidereal  day  ?     86. 

73.  What  is  a  solar  day  ?     87.     Show,  by  means  of  Fig.   13, 
the  reason  of  the  inequality  between  a  sidereal  and  a  solar  day. 

74.  Is  this  inequality  always  the  same  ?     88.     What  is  ap- 
parent time?  mean   time?     What   constitutes   the   civil   day? 
How  is  this  day  divided?     What  constitutes  the  astronomical 
day  ? 

75.  Define  the  equation  of  time.     89.     Why  is  it  sometimes 
added  to  apparent  time,  and  sometimes  subtracted  from  it,  to 
give  the  mean  time?     When  is  its  greatest  additive  value?  its 
greatest  subtractive  value  ?     When  is  the  equation  of  time  zero  ? 

76.  Explain  local  time.     Give  an  example  illustrating  the  prin- 
ciple by  which  the  longitudes  of  places  are  actually  ascertained. 

THE   SEASONS. 

77.  What  do  the  changes  of  the  seasons  result  from  ?     90. 
What  two  circumstances,  already  considered,  regulate  the  supply 
of  heat  which  the  earth  receives  from  the  sun  ? 

78.  Explain  the  effect  which  the  duration  of  the  day  has  upon 
the  supply  of  heat.     91. 

79.  What  second  cause  modifies  the  supply  of  heat  at  a  given 
place  ?     Give  the  quotation.     92. 

So.  Give  the  illustration. 

8 1.  Explain  why  the  hottest  and  coldest  periods  of  the  year 
occur  some  time  after  the  longest  and  shortest  day.     93. 

82.  Does  the  variation  in  the  distance  of  the  earth  from  the 


Il6  NEW    METHOD    OF 

sun,  as  it  performs  its  annual  motion,  have  much  influence  upon 
the  changes  of  the  seasons  ?     94. 

83.  What  is  said  of  the  seasons  viewed  in  connection  with  tbe 
zones  ?     95. 

84.  What  is  said  of  the  seasons  of  the  north  temperate  zone  ? 

85.  Of  the  seasons  of  the  torrid  zone  ? 

86.  Of  the  seasons  of  the  frigid  zones  ? 

87.  Mention  local  and  temporary  causes  which  influence  the 
temperature  of  a  place.     96. 

TWILIGHT. 

88.  Explain  how  the  atmosphere  is  conducive  to  daylight.     97. 
How  it  occasions  the  season  of  twilight. 

89.  Give  the  illustration. 

90.  For  how  long  a  time,  astronomically  considered,  is  twi- 
light generally  said  to  continue  ? 

91.  Upon  what  does  this  time  depend  ?     Illustrate  the  effect 
of  a  change  of  the  angle. 

92.  What  is  said  of  twilight  at  the  equator  ?   at  places  re- 
moved from  the  equator  ? 

93.  Illustrate  upon  the  globe  the  varying  angle  at  which  the 
sun  proceeds  with  regard  to  a  given  horizon  during  the  year. 

94.  When  do  situations  north  of  the  equator  have  their  short- 
est twilight?  when  their  longest?    When  do  southern  situations 
have  their  shortest  and  their  longest  twilight  ? 

95.  When  does  a  parallel  have  morning  and  evening  twilight  ? 
when  a  twilight  lasting  from  sunset  to  sunrise  ?   when  a  con- 
tinuous twilight  ? 

96.  Is  twilight,  ordinarily  considered,  of  the  length  of  astro- 
nomical twilight  ? 


MOUNTING   AND    OPERATING    GLOBES.  I  I/ 


APPENDIX. 


IT  is  not  known  when  globes  were  first  constructed  ;  but  the 
first  celestial  globe  is  supposed  to  have  been  made  by  Anaxi- 
mander  of  Miletus,  a  pupil  of  Thales,  who  flourished  in  the 
sixth  century  before  Christ.  Ptolemy  made  use  of  a  terrestrial 
globe  provided  with  the  universal  meridian,  such  as  is  applied 
to  those  now  in  use.  Tycho  Brahe  had  one  of  copper,  nearly 
five  feet  in  diameter.  One  eleven  feet  in  diameter,  constructed 
at  Limburg,  attracted  the  attention  of  Peter  the  Great,  who  pur- 
chased it,  and  removed  it  to  St.  Petersburg.  It  was  large  enough 
to  accommodate  twelve  persons  sitting  around  a  table  within  it. 
Its  inner  surface  was  celestial,  the  stars  being  represented  by 
gilded  nails;  and  the  outer  surface  was  terrestrial.  In  1851  a 
large  globe  of  novel  construction  was  built  in  Leicester  Square, 
London,  by  Mr.  Wyld.  It  was  fifty-six  feet  in  diameter,  and 
the  delineations  were  upon  the  inside  only.  These  were  modelled 
in  slabs  of  plaster  of  Paris,  which  were  set  like  a  ceiling  on  the 
ribs  of  zinc  which  formed  the  framework  of  the  structure.  The 
slabs  were  cast  in  clay  moulds,  which  were  prepared  with  care 
from  the  most  correct  maps,  on  a  scale  of  ten  miles  to  the  inch. 
About  six  thousand  slabs  were  required  to  cover  the  whole  sur- 
face, their  dimensions  varying  from  two  feet  square  as  the  width 
diminished  towards  the  poles.  The  topographical  features  were 
represented  in  relief,  and  the  surface  painted  in  colors.  A  stair- 
way wound  around  from  the  base,  by  which  the  circular  platforms, 
one  above  another,  were  reached,  that  brought  the  spectators  near 
to  the  inner  surface  of  the  great  shell. 

A  globe  is  made  of  pasted  paper,  eight  or  ten  layers  of  this 
being  applied  successively  to  a  mould  prepared  for  the  purpose. 


Il8  MOUNTING    AND    OPERATING    GLOBES. 

As  this  coating  becomes  dry,  it  shrinks,  and  fits  tightly  over  the 
mould  ;  from  which  it  is  then  removed,  first  being  divided  into 
two  hemispheres.  A  turned  stick  of  right  length,  with  a  short 
wire  in  each  end  for  poles,  is  now  introduced,  one  end  in  each 
hemisphere  ;  and  the  two  shells,  being  brought  together,  are  se- 
cured by  gluing  their  edges.  The  ball  is  now  hung  within  a 
steel  semicircle  just  fitting  its  exterior,  and  coated  with  a  com- 
position of  glue  and  whiting.  Being  made  to  revolve,  the  excess 
of  the  composition  is  removed  by  the  circle  ;  and  the  ball  is  thus 
turned  smooth  and  true,  after  which  it  is  carefully  dried.  The 
next  process  is  to  lay  out  the  lines  of  latitude  and  longitude, 
which  is  done  by  a  beam  compass.  The  maps  are  now  to  be 
cut  into  the  Segments  in  which  they  have  been  engraved ;  and 
these  are  pasted  in  succession  with  white  paste  upon  the  foun- 
dation, or  surface  of  the  ball,  the  meridian  lines  drawn  upon 
this  surface  serving  as  guides.  The  fitting  requires  great  care, 
that  the  edges  may  be  made  to  exactly  coincide  ;  and  some 
stretching  of  the  equatorial  portions  is  sometimes  requisite. 
When  dry,  the  paper  covering  is  colored,  and  then  sized  with 
gelatine,  and  immediately  varnished.  The  final  process  before 
mounting  is  to  dry  again  at  200°  Fahr.  A  globe  is  usually 
covered  with  twenty-six  pieces  of  paper ;  viz.,  two  pole  papers 
or  circles  including  30°  around  each  pole,  and  twenty-four  gores 
meeting  at  the  equator.  Sometimes  the  gores  extend  from  the 
pole  to  the  equator  :  every  gore  has  then  a  narrow  curved  cen- 
tral notch  extending  30°  from  the  equator. 


or  THE 
UNIVERSITY 

OF 


INDEX. 


Altitude 17 

,,       Meridian 17 

Amplitude 17 

Angle 8 

„       Diedral 9 

„       Measurement  of     .     •     .     .    8,  9 

„       Right 9 

Angular  Motion n 

„      Velocity 
Azimuth 

Celestial  Meridian 
Chord 
Circle 

Arc  of 

Day 

Diameter  of 

Division  of  Circumference 

Great 

Hour 

Length  of  Degrees  ....        9 
of  Illumination   .....       27 

Radius  of 8 

Small 11 

„      Vertical 17 

Curve 7 

Day  and  Night,  Alternation  of  .     .       27 
„  „      Effect    on    Length 

of    Approach   to 
the  Poles  ...       33 
,,  ,,      Equal  in  Length,  2S,'3o, 

32 
,,  ,,      Gradual  Change  of 

Length      ...       33 

„  „      of  Frigid  Zones      34, 35 

,,      of  Temperate  Zpnes,    35 

,,      of  Torrid  Zone.     .       35 

„  „      Variable  Length  of,    28, 

29,  3o,  35.  36 

,,  „      When     continuing 

longer    than     24 

Hours  .... 

,,  ,,      When       occurring 

every  24  Hours . 

Day  six  months  everywhere  during 

the  year 36 

Day,  Astronomical 44 

„     Civil 44 

„     Sidereal 42 

„     Solar 43 


.34 


34 


27,  33, 
23,  24, 

22,   28, 


Days  equal  in  Length 

Declination 

Parallels  of     .... 
Earth,  Annual  Motion  of  .... 

Axis  of 12,  25, 

Circumference  of    .... 

Diameter  of 

Distance  from  Sun      .     .     . 
Illumination  of  . 
Orbit  of     .     .     . 
,,      Rotation  of    .     . 
,,      Shape  of  ... 

Ecliptic 

,,         Cardinal  Signs  of    ... 

,,         Divisions  of 

„         Inclination  of     .... 

,,        Plane  of 23, 

Ellipse 

Equator,  Celestial 

„        Division  of  by  Day-Circle, 
,,        Terrestrial      ..... 

Equinoxes 

Globe,  Terrestrial 

Heavenly  Bodies,  Motions  of,   20,  21, 
,,  n         Transit  of 

Hemisphere 

„          Northern 

,,          Southern 

Horizon 

,,       Cardinal  Points  of    ... 

„       Plane  of 

„       Rational 

„       Sensible 

Latitude 

„       Parallels  of 

Line 

„     Straight 

Longitude 

Meridians i^ 

Motion,  difficulty  of  distinguishing 

from  Rest 

Nadir 

Plane      

Point 

Points  of  Compass  at  the  Poles  .     . 

Polar  Circles 13, 

Poles  of  a  Great  Circle  of  a  Sphere, 

„      Ce'estial 

„      Terrestrial 


120 


INDEX. 


Prime  Vertical 17 

Quadrant 8 

Right  Ascension 17 

Seasons,  Beginning  of 32 

Changes  of 45 

Dependence  upon  Zones  .  48 

How  occasioned,  45,  46,  47,  48, 

of  Frigid  Zones  ....  49 

of  Temperate  Zones     .     .  48 

,,       of  Torrid  Zone    ....  49 

Solid 10 

Solstices 32 

Sphere 10 

,,       Celestial 15 

,,       Diameter  of 10 

,,        Oblique 18 

„        Parallel 18 

,,        Radius  of 10 

„        Right 18 

Sun,  Angle  between  Diurnal  Circles 

and  Horizon 40 

,,    Annual  Motion  of     ....  23 

,,    Course  at  North  Pole    ...  37 

„    Course  at  Equator     ....  38 
,,    Course  at  any  Place  between 

the  Equator  and  the  Poles  .  39 

,,     Course  at  Polar  Circles      .     .  40 

„    Course  within  the  Frigid  Zones,  40 

„    Course  at  Spitzbergen  ...  41 


Sun,  Culmination  of 42 

„  Diurnal  Motion  of  ....  37 

,,  Meridian  Altitude  of  ...  41 

„  Motion  between  the  Tropics  .  35 

„  Rate  of  Change  of  Declination,  36 
„  Rising  and  Setting  ...  29,  39 

,,  Spiral  Course  of 37 

,,  Zenith  Distance  of  ....  41 

Surface 7 

,,  Curved 7 

Time,  Apparent 44 

,,  Equation  of 44 

»       Local 44 

„       Mean 44 

,,       Measure  of 42 

Tropics  , 13,  34 

Twil  ght  at  the  Equator     ....  53 
at    Places   removed    from 

the  Equator     ....  53 

Cause  of 51 

Continuance  of    ....  52 

Continuous 54 

lasting  through  the  Night,  54 

Morning  and  Evening .     .  54 

when  longest 54 

when  shortest      ....  54 

Zenith 16 

Zenith  Distance 17 

Zones 13 


Prices  Reduced.  BOSTON,  December,  1877. 

&    HEATH, 


13  Tremont  Place,       ....      BOSTOTT. 


Terms :  Cash  in  Tnirty  Days.  Trade  Price  List. 


ENGLISH. 

A  MANUAL  OF  ENGLISH  LITERATURE,  Histori- 
cal and  Critical :  With  an  Appendix  on  English  Metres.  By  THOMAS  Aa- 
NOLD,  M.  A.,  of  University  College,  Oxford.  American  Edition.  Revised.  LJmo. 
Cloth.  649  pages 

CARPENTER'S     INTRODUCTION     TO     ANGLO- 

SAXON.  An  Introduction,  to  the  study  of  the  Anglo-Saxon  Language,  Com- 
prising an  Elementary  Grammar,  Selections  for  Reading  with  Notes,  and  a 
Vocabulary.  By  STEPHEN  II.  CARPENTER,  Professor  of  Logic  and  English  Litera- 
ture in  the  University  of  Wisconsin,  and  Author  of  "  English  of  the  XIV.  Cen- 
tury." 12mo.  Cloth.  212  pages .  .  . 

CHAUCER'S  PARLAMENT  OP  FOULES.    A  revised 

Text,  witu  Literary  and  Grammatical  Introduction,  Notes,  and  a  full  Glossary. 
By  J.  II.  LOU.NSBURY,  Professor  of  English  in  the  Sheffield  Scientific  School  of 
Yale  College.  12mo.  Cloth.  Ill  pages 

CRAIK'S  ENGLISH  OF  SHAKESPEARE.    Illustrated 

in  a  Philological  Commentary  on  his  Julius  Caesar,  by  GEORGE  L.  CRAIK,  Queen's 
College,  Belfast.  Edited  by  W.  J.  ROLFE,  Cambridge.  16mo.  Cloth.  402 


ELEMENTS    OF    GEOMETRY.    By  GEORGE  A.  WENT- 

\voRTa,  Phillips  Academy,  Exeter. 

ENGLISH  OF  THE   XIV.   CENTURY.     Illustrated  by 

Notes,  Grammatical  and  Etymological,  on  Chaucer's  Prologue  and  Knight's 
Tale.  Designed  to  serve  as  an  Introduction  to  the  Critical  Studv  of  English. 
By  STEPHEN  H.  CARPENTER,  A.  M.,  Professor  of  Rhetoric  and  English  Literature 
in  the  State  University  of  Wisconsin.  12mo.  Cloth.  327  pages 

ESSENTIALS  OF   ENGLISH  GRAMMAR,  for  the 

t'se  of  Schools.  By  Prof.  W.  D.  WHITNEY,  of  Yale  College.  12mo.  Cloth. 
272  |«ges 

This  work  is  founded  on  the  author's  studies  of  language  in  general,  and  on  his 
experience  as  an  examiner  in  English  grammar  for  one  of  the  departments  of  Yale 
College,  and  as  a  teacher  of  French  and  German  in  the  same  institution. 

It  endeavors  to  teach  the  facts  of  English  grammar  in  such  a  wav  as  shall  lay  the 
best  foundation  for  further  and  higher  study  of  language  in  every  department. 

The  work  begins  with  the  formation  of  the  simple  sentence  and  the  distinction  of 
the  parts  of  speech  ;  it  next  explains  the  inflection  of  English  word?,  and  the  forma- 
tion of  derivative  and  compound  words  :  the  parts  of  speech  are  then  taken  up  in 
order  and  treated  in  detail ;  and  chapters  of  syntax  close  the  work.  The  added 
exercises  give  due  opportunity  of  practice  upon  the  matters  taught. 

1 


EITZ'S  CELESTIAL  and  TERRESTRIAL  GLOBES. 

Mounted  and  Operated  by  an  entirely  new  Method,  with  a  Manual  designed  to 
accompany  the  Globe,  the  whole  forming  an  easy  and  practical  introduction  to 
the  study  of  Astronomy  and  Physical  Geography. 

Six-inch  Globe 

Twelve-inch  Globe 

The  Globe,  as  mounted,  makes  such  phenomena  as  the  seasons,  the  changes  in  the 
length  of  day  and  night,  etc..  etc.,  at  once  clear  to  the  apprehension  of  an  ordinary 
child.  For  the  usual  verbal  explanations  which  overtask  the  untrained  imagination, 
it  substitutes  material  surfaces  and  lines  and  real  motions,  and  thus  aids  the  mind 
of  the  learner  in  the  highest  possible  degree  in  forming  correct  and  lasting  impres- 
sions of  phenomena  which,  however  interesting,  are  in  the  abstract  not  easily 
understood. 

FITZ'S  MANUAL.    12mo.   120  pages 

The  Manual  which  accompanies  the  above  Globe  contains  every  direction  for 
its  use,  with  an  explanation  of  the  subjects  which  it  is  specially  designed  to  illustrate, 
such  as  the  changes  in  the  length  of  day  and  night,  the  seasons,  twilight,  etc. 

A  collection  of  more  than  forty  problems  is  also  given,  with  full  directions  for 
working  them,  and  numerous  practical  exercises  to  be  worked  out  by  the  learner  on 
the  Globe.  Questions  for  examination  and  miscellaneous  exercises  are  added  at  the 
end. 

HUDSON'S  FAMILY  SHAKESPEARE :   Plays  selected 

and  prepared,  with  Notes  and  Introductions,  for  Use  in  Families. 
Volume  I.,  containing  As  You  Like  It,  The  Merchant  of  Venice,  Twelfth  Night, 
First  and  Second  of  King  Henry  the  Fourth,  Julius  Cassar,  and  Hamlet. 

Volume  II.,  containing  The  Tempest,  The  Winter's  Tale,  King  Henry  the  Fifth, 
King  Richard  the  Third,  King  Lear,  Macbeth,  and  Antony  and  Cleopatra. 

Volume  III.,  containing  A  Midsummer  Night's  Dream,  Much  Ado  about  Nothing, 
King  Henry  the  Eighth,  Romeo  and  Juliet,  Cymbeline,  Coriolanus,  and  Othello. 
And  Hudson's  Life,  Art,  and  Characters  of  Shakespeare.    2  vols. 

5vols.    Cloth 

Half  morocco 

Full  calf 

HUDSON'S  LIFE,  ART,  AND  CHARACTERS  OF 

SHAKESPEARE.  Including  an  Historical  Sketch  of  the  Origin  and  Growth 
of  the  Drama  in  England,  with  Studies  in  the  Poet's  Dramatic  Architecture, 
Delineation  of  Character,  Humor,  Style,  and  Moral  Spirit,  also  wirh  Critical 
Discourses  on  the  following  plays,  —  A  Midsummer  Night's  Dream,  The  Mer- 
chant of  Venice,  The  Merry  Wives  of  Windsor,  Much  Ado  about  Nothing,  As 
You  Like  It,  Twelfth  Night,  All  ;s  Well  that  Ends  Well,  Measure  for  Measure, 
The  Tempest,  The  Winter's  Tale,  King  John,  King  Richard  the  Second,  King 
Henry  the  Fourth,  King  Henry  the  Fifth,  King  Richard  the  Third,  King  Henry 
the  Eighth,  Romeo  and  Juliet,  Julius  Caesar,  Hamlet,  Macbeth,  King  Lear, 
Antony  and  Cleopatra,  Othello,  Cymbeline,  and  Coriolanus.  In  Two  Volumes. 
12mo.  Cloth.  969  pages 

HUDSON'S  SERMONS.    I6mo.    Cloth. 


HUDSON'S  SCHOOL  SHAKESPEARE.    1st  Series. 

Containing  As  You  LIKE  IT,  THE  Two  PARTS  OP  HENRY  IV., 

THE  MERCHANT  OF  VENICE,  JULIUS  CAESAR, 

TWELFTH  NIGHT,  HAMLET. 

Selected  and  prepared  for  Use  in  Schools,  Clubs,  Classes,  and  Families.    With  In- 
troductions and  Notes.    By  the  REV.  HENRY  N.  HUDSON.     12mo.     Cloth.    636  pages. 

HUDSON'S  SCHOOL  SHAKESPEARE.    2d  Series. 

Containing  THE  TEMPEST,  KING  RICHARD  THE  THIRD, 

THE  WINTER'S  TALE,  KING  LEAR, 

KING  HENRY  THE  FIFTH,                   MACBETH,  ANTONY  AND  CLEOPATRA. 
12mo.     Cloth      678  pages 


HUDSON'S  SCHOOL  SHAKESPEARE.    3d  Series. 

Containing  A  MIDSUMMER  NIGHT'S  DREAM,  ROMEO  AND  JULIET, 

MUCH  ADO  ABOUT  NOTHING,  CYMBELINE, 

KINO  HENRY  VIII.,  OTHELLO.  COBIOLANUS, 

12mo.    Cloth.    666  pages 

2 


HUDSON'S  SEPARATE  PLAYS  OP  SHAKESPEARE. 

THE  MERCHANT  OF  VENICE.     In  Paper  Cover       .  ... 

JULIUS  (LESAR.    In  Paper  Cover  .        .  .  . 

HAMLET.     In  Paper  Cover .  .        . 

THK  TEMPEST.     In  Paper  Cover    ...  .  ... 

MACBETH.     In  Paper  Cover 

HENRY  THE  EIGHTH.     In  Paper  Cover 

AS  YOU  LIKE  IT 

HEXKY  THE  FOURTH.    Parti 

KING  LEAR  

MUCH  ADO  ABOUT  NOTHING . 

ROMEO  AND  JULIET 

OTHELLO . 

A  MIDSUMMER  NIGHT'S  DREAM 

HENRY  THE  FIFTH 

HUDSON'S    PAMPHLET    SECTIONS    OF    TEXT- 

BOOKJS  OF  PROSE  AND  POETRY. 

BURKE.    SECTION  I.    Preface,  Contents,  and  pages  113-218      .... 
BURKE.    SECTION  II.    Life.  Contents,  and  pages  219-325      .... 
.     WEBSTER.     SECTION  I.    Preface,  Contents,  and  pages  335-  421         .        .        . 
WEBSTER     SECTION  II.     Life,  Contents,  and  pages  421  -55§ 

BACON.     Preface,  Life,  Contents,  and  p»ges  553-636 

WORDSWORTH.     SECTION  II.     Preface  and  pages  129  -  251    .... 
COI.EKIDUE  AND  BURNS.     Pref.ice  and  Contents  504 -614     .... 
ADDISON  AND  GOLDSMITH.    80  pages.  


HUDSON'S  TEXT-BOOK  OF  POETRY.  From  WORDS- 
WORTH, COLERIDGE,  BURNS,  BEATTIE,  GOLDSMITH,  and  THOMSON.  With  Sketches 
of  the  Authors'  Lives  For  us*-  in  Schools  and  Classes.  By  the  REV.  H.  N. 
HUDSON.  12mo.  Cloth.  704  pages 

Two  thirds  of  the  volume  is  filled  vrHh  Wordsworth  5  who  has  now  made  good 
his  po.-ition  as  one  of  the  five  great  chiefs  of  English  pot-try,  and  whose  works 
embody,  in  the  highest  form,  whatever  is  noblest  and  best  in  the  English  poetical 
literature  of  the  last  hundred  >ears.  The  whole  of  The  Excursion,  the  first  two 
book"  of  the  Pnlude,  Michael,  The  Brothers,  Tn  tern  dbbey.  Laoilamia  On  the.  Power 
of  Sound,  O'/f.  nn  Innnnrtiiliti;,  and  a  large  number  of  smaller  pieces  are  included. 
The  elections  from  Coleridge  are  The.  jiiictent  Marnier,  Chriatubfl,  Fears  in  Sulttude, 
and  eighteen  other  pieces.  From  Burns,  The  Cotter'*  Sa'urilay  Jfiyht,  The  Tna 
D.JT.S  Tam  0  Shantrr.  The  Vitnn,  and  twenty-six  other  poems,  and  also  twenty- 
nine  of  the  soncrs.  From  Beattie,  The  Minstrel  ,•  from  Goldsmith,  The  Destrfd 
Villa. 'K:  and  from  Thomson,  The  Castle  of  Indolence :  tht-s*  being  generally  regarded 
as  the  be*t  poems  of  those  authors.  In  all  cases,  except  a  few  pieces  of  Words- 
worth's, the  poems  are  given  entire,  as  the  authors  wrote  them. 


HUDSON'S  TEXT-BOOK   OF   PROSE.    From  BURKE, 

WEHSTER,  and  BACON.     With  Notes  and  Sketches  of  the  Authors'  Lives.   By  the 
REV.  H.  N   HUDSON.    12niO.     Cloth     648  pages 

This  work  is  intended  as  a  port  of  coTrranion  volume  to  the  preceding.  It  is  made 
up  on  the  same  general  principle,  the  contents  r.eing  selected  with  a  view  to  unite, 
as  far  a.«  may  be,  the  culture  of  high  and  pure  literary  taste?  with  ihe  attainment 
of  liberal  and  useful  knowledge.  The  selections  from  Burke  are  fatter  to  the  Sher- 
iffs ff  Bristol,  Sprtch  on  Eronomicnl  Rffurin,  Sne.r.rh  to  the  F.lietnr.t  of  Bristol,  and  A 
Lrtft-r  in  n  Wnblt  Lt>r<l,  all  civen  entire  j  also  twenty-nine  other  pieces  gathered,  as 
seemed  best,  from  the  author's  various  works  From  \V'ehster,  Speech  in  Reply  to 
Haitne,  S"fic>i  nn  the  Preniifrnnnl  Pn,tfst,  and  discourse  on  T/ie  Character  ff  Wash- 
inytnn  :  al-o  thirty -two  shorter  pieces  from  oiher  speeches.  From  Bacon  thirty 
of  the  fifty-eight  Essays,  all  given  entire,  and  several  choice  selections  from  Advance* 
ment  t;J  Learning, 


HUDSON'S  CLASSICAL  ENGLISH  READER.   Con- 

taining  selections  from  Bryant,  Burke,  Burns,  Byron,  Carlvle,  Coleridge,  fowley, 
Cowper,  Dana,  Froude,  Gladstone,  Goldsmith,  Grav,  Helps,  Herbert,  Hooker, 
Hume,  Irving,  Keble,  Lamb,  Landor,  Longfellow,  Macnulay,  Milton,  Pea  body, 
Scott,  Shakespeare,  Southey,  Spenser,  Talfourd.  Taylor,  Webster,  Whitrier, 
Wordsworth,  and  other  standard  authors.  With  Explanatory  and  Critical  Foot- 
notes. 12mo.  Cloth.  468  pages 

HALSEY'S  GENEALOGICAL  AND  CHRONOLOGI- 
CAL CHART  of  the  Rulers  of  England,  Scotland,  France,  Germany,  and 
Spain.  By  C.  S.  HALSEY.  Mounted,  33  X  48  inches.  Folded  and  Bound  in  4to, 

10  X  12  inches 

Unbound     

HALSEY'S  BIBLE  CHART  OP  GENEALOGY  AND 

CHRONOLOGY,   from    the    Creation  to  A.  D.   100.    Prepared  by   C.  S. 

HALSEY 

Unbound     .        .        

This  Chart  is  designed  to  illustrate  Bible  History  by  showing  on  a  clear  and  simple 
plan  the  genealogy  and  chronology  of  the  principal  persons  mentioned  in  the  Scrip- 
tures. 

HARVARD  EXAMINATION  PAPERS.    Collected  and 

arranged  by  II.  F.  LEIGHTOX,  A.  M.,  Master  of  Meirose  High  School.  Third 
Edition,  containing  papers  of  June  and  September,  1875-70.  12uio.  Cloth. 

348  pages ... 

These  are  all  the  questions  (except  on  the  subject  of  Geometry),  in  the  form  of 
papers,  which  have  been  used  in  the  examinations  for  admission  to  Harvard  College 
since  1880.  They  will  furnish  an  excellent  series  of  Questions  in  Modern,  Phy.-ical, 
and  Ancient  Geography;  Grecian  and  Roman  Hi.story  ;  Arithmetic  and  Algebra  ; 
Plane  and  Solid  Geometry  ;  Logarithms  and  Trigonometry  ;  Latin  and  Greek  Gram- 
mar and  Composition  ;  Physics  and»Mechanics.  They  have  been  published  in  this 
form  for  the  convenience  of  Teachers,  classes  in  High  Schools,  and  especially  for 
pupils  preparing  for  college. 

OUR  WORLD,  No.  I. ;  or,  First  Lessons  in  Geography. 

Revised  edition,  with  new  Maps,  by  MARY  L.  HALL.     Small  quarto.     119  pages  . 

Designed  to  give  children  clear  and  lasting  impressions  of  the  different  countries  and 

inhabitants  of  the  earth  rather  than  to  tax  the  memory  with  mere  names  and  details. 

OUR  WORLD,  No.  II. ;  or,  Second  Series  of  Lessons 

in  Geography.  By  MARY  L  HALL.  With  fine  illustrations  of  the  various  coun- 
tries, the  inhabitants  and  their  occupations,  and  two  distinct  series  of  Maps, 
5  pages  physical,  and  19  pages  of  finely  engraved  copperplates  political.  Quarto. 

181  pages         

This  book  is  intended,  if  used  in  connection  with  the  First  Lessons,  to  cover  the 
usual  course  of  geographical  study.  It  is  based  upon  the  principle  that  it  is  more 
useful  to  give  vivid  conceptions  of  the  physical  features  and  political  associations  of 
different  regions  than  to  make  pupils  tauiiliar  with  long  lists  of  places  and  a  great 
array  of  statistics. 

OUTLINES  OP  THE  ART  OP  EXPRESSION.    By  J. 

H.  GILMORE,  Professor  of  Logic,  Rhetoric,  and  English  in  the  University  of 

Rochester.     12mo.     Cloth.     117  pages 

This  little  book  was  not  manufactured  to  meet  a  demand,  though  it  is  believed 
that  a  demand  for  such  a  work  has  long  existed.  It  grew,  in  the  author's  class- 
room, out  of  an  attempt  to  remedy  the  det'ec'ive  training  in  English  of  students  who 
were,  in  other  respects,  qualified  to  pursue  to  advantage  a  college  course  Wifh  a 
cursory  survey  of  English  Grammar  (from  the  logical  and  historical  point  of  view) 
it  combines  brief  prartical  suggestions  with  reference  to  Composition  and  Rhetoric, 
together  with  examples  for  criticism  and  topics  for  practice  in  the  construction  of 
sentences. 


PEIRCE'S  TABLES  OP  LOGARITHMIC  and  TRIG- 

OMOMfcXRIG  FU  ACTION  si  TO  THK^iJ  AND  FOUR  ^L,ACi:S  OF 
DECIMALS.  By  JAMES  MILLS  PEIRCE,  University  Professor  of  Mathematics  at 
Harvard  University.  Quarto.  Cloth 

PEIRGE'S  ELEMENTS  OF  LOGARITHMS;  with  an 

Explanation  of  the  Autuor's  THKEt;  AND  FOUR  PLACE  TABLES.  By  JAMES 
MILLS  PURGE,  University  Professor  of  Mathematics  at  Harvard  University. 

12u,o.     Cloth.     92  pages 

This  Work  is  a  Compauion  to  THREE  AND  FOUR  PLACE  TABLES  OF  LOGA- 
B1THMIC  AND  TRIGONOMETRIC  FUNCTIONS,  by  the  same  AutUors. 

STEWART'S    ELEMENTARY    PHYSICS.    American 

Edition.  With  QUESTIONS  and  EXERCISES.  By  PKOF.  G.  A.  HILL,  of  Har- 
vard University.  18uio.  Cloth.  58U  pages 

The  Questions  will  be  direct  and  exhaustive  upon  the  text  of  Mr.  Stewart's  work. 
Af  er  tat;  Questions  will  be  given  a  series  of  easy  Exercises  and  Problems,  designed, 
in  tiiu  li.-mds  of  a  goud  teaciier,  to  arouse  and  strengthen  in  tue  student's  mind  the 
power  of  reasoning  in  accordance  with  sound  scientific  metuods. 

SEARLE'S   OUTLINES   OP  ASTRONOMY.     By  AB- 

THCR  SJARLE,  of  Harvard  College  Observatory,    Ibuio.    Cloth.    433  pages    . 

This  work  is  intended  to  give  such  elementary  instruction  in  the  principal  branches 
of  Astronomy  as  is  required  in  Higu  Schools  or  by  any  students  not  far  advanced  in 
mathematics.  It  is  illustrated  by  carefully  prepared  engravings,  and  contains  some 
information  on  each  of  tlie  following  subjects  :  — 

1-  The  chief  results  of  astronomical  inquiry  up  to  the  present  time  with  regard  to 
the  general  constitution  of  the  universe,  and^  in  particular,  with  regard  to  the  stars, 
planets,  nebulae,  comets,  and  meteors. 

2.  The  methods  of  astronomical  research,  and  their  application  to  the  arts. 

3.  Tiic  general  principles  of  theoretical  astronomy. 

4.  Tue  history  of  astronomy.  • 

5.  Astronomical  statistics. 

THE   ANNALS    OF   THE    ASTRONOMICAL    OB- 

fcERVAToRY  O?1  HARVARD  <JOLL^or£l.  Vol.  VIII.  Results  of  Obser- 
vations made  or  directed  by  WILLIAM  CRANCH  BOND,  A.  M.,  GEORGE  PHILLIPS 
BOND.  A.  M  ,  and  JOSEPH  WINLOCK,  A.  M.  Quarto.  Cloth  .... 

PART   I.  —  Historical  Account  of  the  Observatory  from   October,  1855,  to 
October,  1876. 

PAUT   II.  — T.  Astronomical  Engravings  of  the  Moon,  Planets,  etc. 

II.  Astronomical  Engravings  illustrating  Solar  Phtnomena. 

TH*1  ATLANTIC  PRIMARY  ARITHMETIC.    Simple 

Numbers.     By  G.  L.  DZMAREST      ISmo.    250'  pages 

THE  ELEMENTS  OP  PLANE  TRIGONOMETRY. 

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THE   NATIONAL  MUSIC    COURSE.     In  Four  Books. 

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THE  SECOND  MUSIC  READER.    16mo.    96  pages 

THE  THIRD  MUSIC  READER.    IGmo.    96  pages .        .        .        .      •  .        . 

THE     NATIONAL     MUSIC    CHARTS.      By   LUTHER 

WHITING  MASON.  An  invaluable,  aid  to  Teachers  of  Common  Schools  in  imparting 
a  practica.  knowledge  of  Music,  and  teaching  Children  to  sing  at  sight.  In  Four 
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FIRST  SERIES 

SECOND  SEIIIES  .  

THIRD  SEKIES          ....  . 

FOURTH  SERIES  by  L.  W.  MASON  and  J.  B  SHARLAND 
EASEL 


THE    NATIONAL   MUSIC    TEACHER.      A   Practical 

Guide  for  Teaching  Vocal  Music  to  Young  Children.     By  L.  W.  MASON.     8vo. 
72  pages  

GERMAN. 

A  COURSE  IN  SCIENTIFIC  GEBMAN.    Prepared  by 

H.  B.  HODGES,  Instructor  in   Chemis'ry  and  German  in  Harvard  University. 
With  Vocabulary.     12uio     Cloth.     182  pages 

This  book  has  been  prepared  to  supply  a  want  long  felt  '>y  English  and  American 
ptudents  of  science,  of  some  aid  in  the  acquirement  of  a  knowledge  of  the  German 
language  of  a  sufficiently  practical  nature  to  enable  them  to  read  with  ease  the  scien- 
tific literature  of  Germany. 


GEEEK. 

GOODWIN'S   GEEEK  GRAMMAR.    By  WILLIAM  W. 

GOODWIX,  Ph.  D.,  Eliot  Professor  of  Greek  Literature  in  Harvard  University. 

12mo.     Half  morocco.     262  pages 

The  object  of  this  Grammar  is  to  state  general  principles  clearly  and  distinctly, 
with  special  regard  to  those  who  are  preparing  for  college.  In  the  sections  on  the 
Moods  are  stated,  for  the'first  time  in  an  elementary  form,  the  principles  which  are 
elaborated  in  detail  in  the  author's  "  Syntax  of  the  Greek  Moods  and  Tenses.;' 


GBEEK    MOODS    AND    TENSES.     The  Sixth  Edition. 

By  WILLIAM  W.  GOODWIN,  Eliot  Professor  of  Greek  Literature  in  Harvard  Uni- 
versity.    1  vol.    12mo.     Cloth.    264  pages 

This  work  was  first  published  in  1860,  and  it  appeared  in  a  new  form  —  much  en- 
larged and  in  great  part  rewritten  —  in  1865.  In  the  present  edition  the  whole  has 
been  again  revised  ;  some  sections  and  notes  have  been  rewritten,  and  a  few  notes 
have  been  added.  The  object  of  the  work  is  to  give  a  plain  statement  of  the  princi- 
ples which  govern  the  construction  of  the  Greek  Moods  and  Tenses,  —  the  most  im- 
portant and  the  most  difficult  part  of  Greek  Syntax. 

GOODWIN'S   GBEEK   READER.     Consisting  of  Extracts 

from  Xenophon,  Plato,  Herodotus,  and  Thucvdides  ;  being  the  full  amount  of 
Greek  Prose  required  for  admission  at  Harvard.     With  Maps,  Notes,  References 
to  GOODWIN'S  GREEK  GRAMMAR,  and  parallel  References  to  CROSBY'S  and 
HADLEY'S   GRAMMARS      Second  edition,  edited  by  PROFESSOR  W.  W.  GOOD- 
WIX, of  Harvard  College.     12mo.     Half  morocco.    334  pages       .... 
The  revised  edition  contains  the  first  and  second  books  of  the  Anabasis  (in  place 
of  the  third  and  fourth  books  of  the  former  editions)  with  copious  notes,  the  greater 
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first  chapter  of  the  Memorabilia,  of  Xenophon  ;  the  last  part  of  the  Apology,  and  the 
beginning  and  end  of  the  Phaedo,  of  Plato  ;  selections  from  the  sixth,  seventh,  and 
eighth  books  of  Herodotus,  and  from  the  fourth  book  of  Thucydides. 


GOODWIN'S    SELECTIONS    PROM    XENOPHON 

AND  HERODOTUS.  With  Notes  adapted  to  Goodwin's  Greek  Grammar, 
Parallel  References  to  Crosby's  and  Hadley's  Grammars,  and  copper-plate 
Maps.  Edited  by  PROFESSOR  W.  W.  GOODWIX  and  Joax  WILLIAMS  WHITE,  of 

Harvard  College.     12tno.     Half  morocco     408  pages 

This  book  contains  the  first  four  books  of  the  Anabasis,  and  the  greater  part  of 
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and  eighth  books  of  Herodotus.  It  has  been  prepared  for  the  use  of  those  who  from 
want  of  time  or  for  other  reasons  are  unable  to  read  the  greater  variety  of  selections 
in  Greek  Prose  which  are  contained  in  Goodwin's  Greek  Reader. 


THE  FIRST  FOUR  BOOKS  OF  THE  ANABASIS 

OF  XENOPHON.  Edited,  with  copious  Notes  and  References  to  Goodwin's 
Greek  Grammar,  Parallel  References  to  Crosby's  and  Hadley's  Grammar* ,  and 
a  copper-plate  Map,  by  PROFESSOR  W.  W.  GOODWIX  and  JOHX  WILLIAMS  WH.TB, 
of  Harvard  College.  12mo.  Half  morocco.  240  pages 

LEIGHTON'S  GREEK  LESSONS.  Preparedto accompany 
Goodwin's  Greek  Grammar.     By  R.  F.  LEIG;JTON,  Master  of  Melrose  High  School. 

12mo.     Half  morocco.     264  pages 

This  work  contains  about  one  hundred  lessons,  with  a  progressive  perils  of  exer- 
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Anabasis.  The  exercises  on  the  Moods  are  sufficient,  it  is  believed,  to  develop  the 
general  principles  as  stated  in  the  Grammar.  The  text  of  four  chapters  of  the 
Anabasis  is  given  entire,  with  notes  and  references.  Full  vocabularies  accompany 
the  book. 


LIDDELL   &   SCOTT'S   GKEEK-ENGLISH   LEXI- 

CON.  Abridged  from  the  new  Oxford  Edition.  New  Edition.  With  Appendix 
of  Proper  and  Geographical  Names,  bj  J.  M.  WHITON.  Square  12mo.  835  pages. 

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The  English  editions  of  Liddell  &  Scott  are  not  stereotyped ;  but  each  has  been 
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is  larger  by  one  eighth  than  the  fifth,  and  contains  1,865  pages.  It  is  an  entirely  dif- 
ferent work  from  the  first  edition,  the  whole  department  of  etymology  having  been 
rewritten  in  the  light  of  modern  investigations,  and  the  forms  of  the  irregular  verbs 
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first  time  brought  within  the  means  of  the  great  body  of  American  scholars. 

PLATO'S  APOLOGY  OF  SOCRATES  AND  CRITO. 

Edited  by  JOHN  WILLIAMS  WHITE,  Ph.  D.,  Assistant  Professor  of  Greek  in 

Harvard  University. 

The  basis  of  this  work  will  be  the  German  edition  of  Dr.  Christian  Cron.  (Platons 
Vertheidigungsrede  des  Sokrates  und  Kriton.  Sechste  Auflage.  Leipzig,  Teubner, 
1875.)  To  the  matter  contained  in  Dr.  Crou's  edition  there  will  be  added  notes  by 
the  Editor  and  from  other  sources,  analyses,  and  extended  references  to  Goodwin 
and  Hadley.  The  book  will  be  for  the  class-room,  and  all  matter  not  of  direct  value 
to  the  student  will  be  rigidly  excluded. 

THE  CEDIPUS  TYRANNUS  OP  SOPHOCLES.    Ed- 

ited,  with  an  Introduction,  Notes,  and  full  explanation  of  the  metres,  by  JOHN 
WILLIAMS  WHITE.  Ph.  D. ,  Assistant  Professor  of  Greek  in  Harvard  University. 
12mo.  Cloth.  219  pages 

THE  MEDEA  OP  EURIPIDES.  Edited,  with  Notes  and 
an  Introduction,  by  FREDERIC  D.  ALLEN,  Ph.  D.,  Professor  in  the  University  of 
Cincinnati.  12mo.  Cloth.  141  pages 

SIDGWICK'S       INTRODUCTION      TO      GREEK 

PROSE  COMPOSITION.    12mo.    Cloth.    280  pages 

WHITE'S   FIRST  LESSONS   IN    GREEK.    Prepared 

to  accompany  Goodwin's  Greek  Grammar,  and  designed  as  an  Introduction  to 
his  Greek  Header.    By  JOHN  WILLIAMS  WHITE,  Ph  D.,  Assistant  Professor  of 
Greek  in  Harvard  University.     12mo.    Half  morocco.    305  pages 
A  series  of  seventy -five  lessons  with  progressive  Greek-English  and  English-Greek 

exercises.     Followed  by  a  series  of  additional  exercises  on  Forms,  and  complete 

vocabularies. 

WHITON'S  SELECT  ORATIONS  OF  LYSIAS.  Com- 
prising the  Defence  of  Mantitheus,  the  Oration  against  Eratosthenes,  the 
Keply  to  u  The  Overthrow  of  the  Democracy,"  and  the  Areopagitic  Oration 
concerning  the  Sacred  Olive-Trunk.  Edited  by  JAMES  MORRIS  WHITON,  Ph.  D. 

12mo.  151  pages 

The  grammatical  notes  deal  almost  wholly  with  the  syntax,  —  as  befits  a  work  of 

this  grade,  —  and  have  been  prepared  with  a  vspecial  aim  to  elucidate  the  usage  of 

the  verb.     References  are  made,  for  the  most  part,  to  Goodwin's  Greek  Moods  and 

Tenses,  and  Goodwin's  and  Hadley 's  Grammars. 

YONGE'S     ENGLISH-GREEK    LEXICON.      Square 

12mo.    Cloth.    483  pages 


LATIN. 

ALLEN  &  GREENOUGH'S  LATIN   GRAMMAR: 

a  Latin  Grammar  for  schools  and  colleges,  founded  on  Comparative  Grammar. 
12mo.  Half  morocco.  290  pages  (including  supplementary  Outlines  of  Syntax, 
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The  features  of  this  grammar  to  which  we  invite  attention,  are  :  1.  The  scientific 
form  of  statement  in  the  Etymology,  corresponding  to  the  most  advanced  views  of 
comparative  philologists  ;  2.  The  comparison  with  kindred  languages  suggested 
throughout,  especially  in  numerous  brief  philological  notes,  and  in  references  to  the 
syntax  of  Goodwin's  Greek  Grammar  ;  the  grouping  and  subordination  of  topics  in 
the  Syntax,— which  contains  nearly  200  cross-references,  with  upwards  of  1,000 
citations  Irom  classic  authors,  —  so  that  unusual  brevity  is  attained  without  sacri- 
fice of  completeness. 

ALLEN  &   GREENOUGH'S  LATIN  METHOD:   a 

Method  of  Instruction  iu  Latin  ;  being  a  companion  and  guide  in  the  study  of 
Latin  Grammar.  With  elementary  instruction  in  Heading  at  Sight,  Exercises 
in  Translation  and  Writing,  Notes,  and  Vocabularies  ;  also  "  Outlines  of  Syn- 
tax," taken  from  the  Latin  Grammar.  12mo.  Cloth.  134  pages  . 

ALLEN  &  GREENOUGH'S  LATIN  COMPOSITION: 

an  Elementary  Guide  to  Writing  in  Latin.    Part  I.  Constructions  of  Syntax; 

Part  II.  Exercises  in  Translation.  12mo.  Cloth.  198  pages  .... 
Part  First  (which  is  published  separately)  consists  of  thirty  progressive  Lessons, 
with  full  instructions,  exercises,  and  vocabulary  ;  and  is  designed  "  to  furnish  a 
sufficient  amount  of  study  and  practice  in  Latin  composition  during  the  last  year 
of  preparation  for  college,  or  the  first  of  a  college  course."  Part  Second  consists  of 
about  forty  exercises  in  translation,  chiefly  narrative,  adapted  to  the  use  of  advanced 
or  college  classes  ;  with  annotated  references  to  the  Lessons  of  Part  I.,  and  to  the 
sections  of  a  special  Introduction  on  the  Choice  of  Words,  the  Form  of  the  Sentence, 
and  Idiomatic  Usages. 

ALLEN  &    GREENOUGH'S   CJESAR:    Csesar's  Gallic 

War:  Four  Books.  With  Historical  Introduction,  Notes,  and  a  Copperplate 
Map  of  Gaul.  [With  a  full  Vocabulary  by  R.  F.  PENNELL,  of  Phillips  Exeter 

Academy.]    12mo.    Half  morocco.    282  pages 

Without  Vocabulary  ...  

The  text  of  this  edition  is  that  of  Nipperdey,  important  variations  being  noticed. 
The  notes  are  unusually  full  in  historical  illustration,  derived  largely  from  Momm- 
Ben,  Long,  Merivale,  the  "History  of  Julius  Caesar"  by  Napoleon  III.,  and  the 
excellent  school  edition  of  Moberly.  In  the  earlier  portions  they  are  especially  de- 
signed to  guide  in  a  systematic  and  careful  study  of  Latin  syntax. 

ALLEN  &  GREENOUGH'S  SALLUST:  The  Conspiracy 

of  Catiline  as  related  by  Sallust.  With  Introduction  and  Notes  explanatory 
and  historical.  12mo.  Cloth.  84  pages 

ALLEN  &  GREENOUGH'S  CICERO:  Select  Orations  of 

Cicero,  chronologically  arranged,  cover'ng  the  entire  period  of  hi*  public  life. 
From  the  text  of  Baiter  and  Kayser.  With  Life,  general  and  special  Introduc- 
tions, and  Index  of  topics  discussed.  12mo.  Half  morocco.  394  pages  . 

The  text  without  notes 

It  is  the  design  of  this  edition  to  give  a  full  view  of  Cicero's  public  career,  as  ora- 
tor and  statesman,  extending  through  about  forty  of  the  most  eventful  years  of  the 
later  Republic.  With  this  view,  the  selection  includes  the  earliest  and  the  latest 
of  his  public  orations,  while  the  special  Introductions  cover  very  fully  the  inter- 
vening political  history.  Besides  the  orations  more  commonly  read  in  schools,  are 
given  the  Roscius  and  Sestius  (ahri  Jged),  with  the  first  against  Verres  and  the  last 
Philippic,  —  thirteen  in  all, —  with  one  or  two  short  passages  of  special  celebrity ,  for 
practice  in  reading  at  sight.  Especial  care  has  been  taken  in  the  departmene  of 
Antiquities,  which  has  been  treated  in  numerous  notes  (in  smaller  type),  some  of 
them  —  as  that  on  the  Homan  Aristocracy  —  being  brief  essays  on  the  several  topics. 
The  Introduction  contains  a  classified  list  of  all  the  works  of  Cicero,  with  the 
occasions  and  topics  of  all  of  his  orations. 

9 


ALLEN  &  GREENOUGH'S  CATO  MAJOR:    Cicero 

De  Senectute,  a  Dialogue  on  Old  Age.  With  Introduction  (on  the  adoption  in 
Rome  of  the  Greek  philosophy)  and  Notes.  12mo.  Cloth.  57  pages 

\ 

ALLEN  &  GREENOUGH'S  OVID :   Selections  from  the 

Poems  of  Ovid,  chiefly  the  Metamorphoses.  With  special  Introductions,  Notes, 
and  Index  of  Proper  Names.  12mo.  Half  morocco.  283  pages 

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selections  include  those  of  special  interest  as  illustrating  the  poet's  life  ;  and  a  lisfc 
Is  given  of  all  his  writings,  with  their  topics  and  occasions.  The  Notes  contain 
brief  instructions  on  scanning  at  sight. 

ALLEN  &  GREENOUGH'S  VIRGIL :    The  Poems  of 

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wealth  of  traditional  interpretation  which  is  more  important,  perhaps,  in  the  study 
of  Virgil  than  in  that  of  any  other  ancient  poet." 

ALLEN  &   GREENOUGH'S  VIRGIL.    With  Vocabu- 
lary.   12mo.    Half  morocco.    588  pages 

ALLEN  &  GREENOUGH'S  Course  No.  I.    A  Full  Pre- 

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TUTE  ( Cato  Major).  12ino.  Half  morocco.  582  pages 

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Half  morocco.  518  pages 

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leges. 

ALLEN  &   GREENOUGH'S  CICERO.    Eight  Orations 

of  Cicero.    With  Vocabulary  by  R.  F.  PENNELL.    12mo.    Half  morocco.    358 


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(Jugurtha,  abridged),  Ovid,  Virgil,  Plautus,  and  Terence  (single  scenes),  Cicero 
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10 


n» 

^ _. 


YB   1712! 
YB   17122 


G1N.N    AND    HEATH, 

13  TREMONT  PLACE,  BOSTON,  MASS, 


SHAKESPEARES. 

Craik's  English  of  Shakespeare. 
Hudson's  Family  Shakespeare.  5vols. 
Hudson's  Life,  Art,  and  Character  of 

Shakespeare.    2  vols. 
Hudson's  School  Shakespeare.     1st. 

2d,  and  3d  Series. 

PAMPHLET  EDITION  OF 

Hudson's  "Wordsworth. 
Coleridge. 
Burns. 
Burke. 
Webster. 
Bacon.' 
Beattie. 
Goldsmith. 
Thomson.    And 

SHAKESPEARE'S 

Merchant  of  Venice. 

Julius  Caesar. 

Hamlet.  Othello. 

Tempest. 

Macbeth. 

Henry  VIII.        Henry  V. 

As  You  Like  It. 

King  Lear. 

King  Henry  IV.    1st  Fart. 

Much  Ado  about  Nothing. 

Romeo  and  Juliet. 

Midsummer  Night's  Dream. 

GEOGRAPHIES. 

Our  World,  No.  1;    or,  First  Lessons  in 

Geography. 
Our  World,  No.  II. ;  or,  Second  Series 

of  Lessons  in  Geography. 

MUSIC. 

The  National  Music  Coiuse.  By 
Messrs.  Mason,  Kichber^,  Sharland, 
and  Holt. 

Primary  or  First  Music  Reader. 

Intermediate  Music  Reader. 

Fourth  Music  Reader. 

Fifth  or  High  School  Music  Reader. 

Second  and  Third  Music  Readers. 

1st,  2d,  3d,  and  4th  Series  Music 
Charts. 


JKj          Vsllclj 


MISCELLANEOUS 

Arnold's  English  Literature. 
English  of  the  XIV.  Century. 
Fitz's  Globe  and  Manual. 
Gilmore's  Art  of  Expression. 
Harvard  Examination  Papers. 
Hudson's  Text-Book  of  Poetry. 
Hudson's  Text-Book  of  Prose. 
Peirce's  Tables  pf  Logarithms. 
Peirce's  Elements  of  Logarithms. 
Wentworth's  Geometry. 
Wheeler's  Trigonometry. 
Whitney's  Essentials  Eng.  Grammar 

GREEK. 

Goodwin's  Greek  Grammar. 

Goodwin's  Greek  Moods  and  Tenses. 

Goodwin's  Greek  Reader. 

Goodwin's  Selections  from  Anaba- 
sis and  Herodotus. 

White's  First  Lessons  in  Greek. 

White's  CEdipus  Tyrannus  of  Sopho- 
cles. 

Whiton's  Select  Orations  of  Lysias. 

Leighton's  Greek  Lessons. 

Liddell  and  Scott's  Greek-English 
Lexicon.  14th  edition. 

Liddell  and  Scott's  Greek-English 
Lexicon.  The  Sixth  Oxford  edition, 
unabridged.  4to. 

Allen's  Medea  of  Euripides. 

LATIN. 

Allen  &  Greenough's  Latin  Grammar. 

Latin  Method. 

Latin  Composition. 

Caesar. 

Cicero. 

Ovid. 

Virgil. 

Sallust. 

Cato  Major. 

COURSE  No.  I.— (Without  Vocabulary),  con- 
taining four  books  of  Csesar,  Sallugt's  Catiline, 
eight  Orations  of  Cicero,  ami  the  Cato  Major. 

COUKSI-:  No.  11  —  (With  Vocabulary),  con- 
taining four  books  of  Caesar's  Gallic  War,  and 
eight  Orations  of  Cicero. 

Leighton's  Latin  Lessons. 

Allen's  Latin  Lessons,  Lexicon,  Prim- 
er, Grammar,  Reader,  and  Compo- 
sition. 

Madvig's  Latin  Grammar. 

White's  Junior  Student's  Complete 
Latin-English  Lexicon. 


